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Real-world use of angiotensin-converting enzyme inhibitors/angiotensin receptor blockers/β-blocks in Chinese patients before acute myocardial infarction occurs: patient characteristics and hospital follow-up
Xuhe Gong ORCID: orcid.org/0000-0002-0513-66741,
Xiaosong Ding1,
Hui Chen1 &
Hongwei Li1,2,3
Current guidelines recommend angiotensin-converting-enzyme inhibitors (ACEI) or angiotensin-receptor blockers (ARB) or β-blockers (β-B) for secondary prevention in patients after an acute myocardial infarction (AMI). However, there is limited data to evaluate ACEI/ARB/β-B (AAβ) used before AMI on major adverse cardiovascular events (MACE), in China patients.
This study sought to investigate whether AAβ treatment prior to AMI is associated with better hospital outcomes at the onset of AMI.
A total of 2705 patients were selected from the Cardiovascular Center Beijing Friendship Hospital Database Bank, and divided into two groups on the basis of admission prescription: AAβ (n = 872) or no-AAβ (n = 1833). The study was also designed using propensity-score matching (226 AAβ treated patients vs 452 no-AAβ treated patients). The primary outcome was a composite of cardiac death and heart function and infarct size during hospitalization follow-up.
The mean follow-up period was about 8 days in MACE. The Cox model showed the two groups had similar risk of cardiac death. The in-hospital mortality was 3.36% (3.33% of AAβ users and 3.38% of nonusers, p = 0.94). In adjusted analysis, there was still no difference in in-hospital mortality between the two groups (3.54% vs 2.88%, p = 0.64). However, the AAβ treated patients were associated with better heart function and smaller infarct size than the no-AAβ treated patients.
The in-hospital MACE was similar between AAβ treated patients and no-AAβ treated patients. However, treatment with AAβ before AMI was associated with improved heart function and smaller infarct size.
Ischemic heart disease is one of the most frequent diseases worldwide; and cardiovascular diseases are among the leading causes of death in developed industrial countries. With the development of China's economy, the number of patients with acute myocardial infarction (AMI) increases year by year in China, and the overall mortality rate is on the rise [1]. Although interventional therapy has greatly improved the prognosis of myocardial infarction, the basic drug therapy is also essential. A large number of clinical trials have found that angiotensin-converting enzyme inhibitors (ACEIs), Angiotensin-receptor blockers (ARBs) and β-blockers (β-B) prevented ischemic events and mortality in patients with AMI [2,3,4]. Thus, secondary prevention protocols including these agents are regarded to be standard therapy following an AMI, along with aspirin and statins [5, 6]. Although there is no doubt that ACEI/ARB/β-blocker (AAβ) offer the most benefit to AMI patients, there is still uncertainty about prescribing these agents to a real population of patients before AMI occurs.
Moreover, several previous studies have documented less benefit with these agents in patients with lower-risk myocardial infarctions [7, 8]. Was it possible to improve the prognosis by taking so many drugs before myocardial infarction occurs? This is a question. In addition, most studies have a longer follow-up time, and we only focus on events during hospitalization. Therefore, by using Cardiovascular Center Beijing Friendship Hospital Database Bank, we sought to evaluate the effectiveness of AAβ treatment in improving hospital survival. This study focused on the left ventricular functions evaluated by echocardiography, myocardial infarct size estimated by peak concentration of myocardial enzyme and the major cardiovascular events (MACE) in hospital, the MACE includes cardiac-death, target vascular reconstruction, recurrent myocardial infarction, malignant arrhythmia, cerebral infarction and cerebral hemorrhage.
The present study was based on the Cardiovascular Center Beijing Friendship Hospital Database Bank (CBD Bank). Briefly, this is a single center study. From January 2013 to October 2016, a total of 2712 consecutive patients with AMI were enrolled in this study. The local institutional review board at our hospital approved the study protocol, and this study was in accord with the Declaration of Helsinki.
Inclusion criteria for the present analysis were as follows: (1) consecutive patients 18 years of age or older; (2) patients diagnosed with ST-segment elevation AMI (STEMI) or non-ST-segment elevation AMI (NSTEMI). Exclusion criteria were (1) a lack of documentation of prescribed medications on admission; (2) both ARB and ACEI received; (3) infectious diseases (tuberculosis, active infective endocarditis), rheumatic disease (systemic lupus erythematosus, rheumatoid arthritis, vasculitis), hematological diseases (leukemia, lymphoma, disseminated intravascular coagulation) and neoplastic disease.
Finally, a total of 2705 patients were included in this study, the study was also designed using propensity-score matching to assemble a balanced cohort. The patient flow of the study is shown in Fig. 1.
Flow chart of patient enrollment. AAβ angiotensin-converting-enzyme inhibitors (ACEI) or angiotensin-receptor blockers (ARB) or β-blockers (β-B), AMI acute myocardial infarction, MACE major adverse cardiovascular events
The basic characteristics data
The hospital medical records were detailed and intact. Most of the data was extracted from the medical records including demographic data (age and sex), history of past illness (hypertension, coronary disease, diabetes, hyperlipemia and other diseases), conditions of smoking and drinking, family histories [hypertension, diabetes and coronary heart disease (CHD)] and medications (ACEI, ARB, β-blocker and other) before admission. Body mass index (BMI) was calculated by dividing weight in kilograms by height in meters squared (kg/m2).
We analyzed baseline demographic characteristics, history of past illness, initial laboratory test results and medications. Blood samples for baseline laboratory tests were collected at admission or during the first 5 days after presentation of acute myocardial infarction. Serum peak concentration of cardiac troponin I (cTnI), myoglobin (Myo), creatine kinase-myocardial band (CKMB) level were used for estimation of infarct size. The LV ejection fraction was determined using 2-dimensional echocardiography during the index hospitalization. In-hospital complications and their management were also recorded.
The major adverse cardiac events (MACEs) in hospital were defined as cardiac death, target vascular reconstruction, malignant arrhythmia, recurrent myocardial infarction, cerebral infarction and cerebral hemorrhage.
Continuous variables are presented as mean ± standard deviations or median with interquartile range, and were compared using the unpaired Student's t tests or the Mann–Whitney U test. Categorical variables are expressed as frequencies and percentages, and were compared by Chisquare or Fisher's extract statistics. Patients were categorized into two groups: patients receiving AAβ, and patients no receiving AAβ. Since patients were not randomly assigned to AAβ or no-AAβ, 1:2 propensity score (PS) matching based on their probability of using AAβ was performed to reduce the effect of treatment-selection bias and potential confounding factors in this observational study. For each patient, a PS indicating the likelihood of using AAβ before hospitalization was calculated using a non-parsimonious multivariable logistic regression model with covariates including baseline demographic characteristics, such as age, sex and body mass index (BMI), past medical history including hypertension, diabetes mellitus, hyperlipidemia, heart failure, renal dysfunction and coronary heart disease.
Survival curves were conducted using Kaplan–Meier estimates and compared with the log-rank test. The multivariate Cox proportional hazards regression analysis was used to assess the association between adverse clinical events and the AAβ and no-AAβ groups. All factors showing significance in the univariate analysis (p < 0.05) were then examined by a multivariate analysis.
All statistical tests were two-tailed, with statistical significance defined as a p value of < 0.05. All analyses were performed by using SAS 9.4 (SAS Institute Inc., Cary, NC, USA) and Metaninf function in Stata 12.0.
Overall population
Among the 2705 eligible patients, AAβ were prescribed to 872 patients (32.2%), and no-AAβ group was 1833 patients (67.8%) pre-admission. The median age was 65 years (interquartile range 56–77 years); 70.9% of the patients were men. A total of 33.9% of the patients had diabetes, 65.4% had hypertension, and just 1.29% of patients had LV systolic dysfunction. The baseline clinical characteristics are shown in Table 1.
Table 1 Baseline characteristics
Significant correlates of AAβ therapy in multivariable analysis are shown in Fig. 2. Compared with no AAβ-treated patients, patients prescribed AAβ prior to admission were more likely to be women, and had worse baseline clinical: the higher BMI, the higher systolic and diastolic blood pressure at admission; what's more, the proportion of hypertension, diabetes, heart failure, stroke, old myocardial infarction and renal insufficiency is higher. Also, the no AAβ-treated patients were more likely to have acute ST segment elevation myocardial infarction (STEMI); hence the proportion of emergency PCI (percutaneous coronary intervention) is higher.
Factors associated with AAβ use in multivariable analysis. Variables associated with AAβ use are shown along the vertical axis. The strength of effect is shown along the horizontal axis with the vertical line demarcating an odds ratio (OR) of 1 (i.e., no association); estimates to the right (i.e., > 1) are associated with a greater likelihood of AAβ use, whereas those to the left (i.e., < 1) indicate a reduced likelihood of AAβ use. Each dot represents the point estimate of the effect of that variable in the model, whereas the line shows the 95% confidence interval (CI)
Propensity score-matched population
Propensity scores for AAβ use, calculated for 678 patients, were used to match 226 patients receiving AAβ (33.3%) with 452 patients no receiving AAβ (66.7%). There were no significant differences in baseline clinical, past medical history, types of myocardial infarction between the AAβ treated and no- AAβ treated patients for the propensity score-matched subjects, except for previous PCI (p = 0.02, Table 1).
The estimated infarction size and left ventricular function between AAβ treated patients and no-AAβ treated patients.
Serum peak concentration of cTnI, Myo and CKMB level were used for estimation of infarct size. We found no difference in pMyo between the two groups, however, there was higher peak level of serum myocardial enzymes (p-CKMB and p-cTnI) in the no-AAβ treated patients. (p-CKMB: 57.0 vs 31.9, p = 0.02, p-cTnI: 6.0 vs 3.0, p = 0.002, Table 2).
Table 2 The estimated infarction size between AAβ and NO-AAβ group
From the perspective of cardiac function assessed by echocardiography, the AAβ treated patients were associated with better heart function and smaller infarct size than the no-AAβ treated patients. In terms of cardiac function evaluation, the left ventricular ejection fraction (EF) and fraction shortening (FS) in the AAβ treated patients were significantly higher than the no-AAβ treated patients. (EF: 0.63 vs 0.61, p = 0.009, FS: 0.34 vs 0.33, p = 0.004, Table 3).
Table 3 The comparison of left ventricular function between AAβ and no-AAβ group
The median follow-up duration was 8 days (interquartile range 6–10). All-cause death occurred in 91 patients (3.4%) in the overall population. There were no significant associations between the treatment strategy and all-cause death or cardiac death. The multivariate Cox proportional hazards regression analysis showed both the AAβ treated patients and no-AAβ treated patients had similar risk of cardiac death or all-cause death (cardiac death, p = 0.72, all-cause death, p = 0.94, Table 4).
Table 4 Comparison of clinical outcomes during hospitalization between study groups
After propensity-score matching, all-cause death occurred in 3.5 and 2.9% of matched patients receiving AAβ and no-AAβ, no significant differences were also observed in the incidence of all-cause death between the two groups.
In survival analysis, in-hospital death was no significant differences between the two groups. After adjusting for baseline clinical and propensity score, there were also no significant differences (Fig. 3).
Survival curves of all-cause death during hospitalization between the groups. a Overall population. b Propensity score-matched population
In this single center observational study, we found that previous treatment with AAβ was associated with a non-significant reduction in the risk of all-cause mortality during hospitalization. However, previous use of AAβ reduced myocardial infarction size and improved heart function.
The clinical benefit of AAβ in patients after AMI may be partly mediated by a reduction in the risk of recurrent ischemic events and reduction in congestive heart failure [9, 10], some studies emphasizing that the AAβ have an additive effect, which have long been reflected in the clinical guidelines, which recommend routine use of AAβ in all AMI patients [11, 12]. However, Could the drugs used in the past improve the condition of myocardial infarction? Moreover, it is controversial whether this medical prevention improves clinical outcomes in hospital. Therefore, the adherence to these guideline-based medications differs substantially among cardiovascular societies [13, 14]. Thus, considering the potential adverse events attributed to over-use of AAβ treatment prior to AMI, treatment of the unselected population with these agents might be inappropriate in the modern PCI era [15]. Our current study, based on reliable data that included all AMI patient from 2013 to 2016, may provide an important ''real world'' insight into this debatable issue.
In our study population, 11.9% of patients had a history of myocardial infarction; we found that 53.6% of patients with old myocardial infarction were not taking the AAβ, which suggests that application of medicine for improving prognosis of myocardial infarction was still inadequate. A recent clinical research conducted by Liu et al. [16] that focused on the use and trends of ACEI/ARB therapy in China over the past decade (2001–2011), after analyzing 102,003 patients, they found that one-third of Chinese AMI patients with Class I indications do not receive ACEI/ARB therapy during hospitalization, with little improvement in rates over time. The underutilization of ACEI/ARB therapy was also observed in our study. Moreover, the main characteristics of patients who are more willing to take AAβ are as follows: the history of hypertension, coronary heart disease and heart failure.
In our present analysis, the all cause death was no significant difference in patients who were treated with either AAβ or not before AMI occurs. After matching, there was also no difference in mortality between the two groups. This finding may be partly explained by the short follow-up time, we only observed the deaths in the hospital. If we extend the follow-up period, for example, such as 1 year, 2 years, or longer, there may be a benefit of AAβ treatment.
Moreover, could the AAβ improve the MACE? This research found that the treatment strategy was not related to target vascular reconstruction, malignant arrhythmia, cerebral infarction and cerebral hemorrhage. The only difference was the proportion of recurrent myocardial infarction, which was reduced in the no-AAβ treated patients (2.47% vs 4.48%, p = 0.005); However, this difference disappears after matching. Therefore, the AAβ did not improve the in-hospital MACE. Similarly, as the follow-up time increases, the role of AAβ therapy in improving MACE may be apparent.
Although the hospital mortality was no difference between the two groups, the use of AAβ really reduced the size of the infarct area and improved heart function; In other words, the drugs used in the past improved the condition of myocardial infarction. The serum peak concentration of Myo, cTnI, CKMB level was used for estimation of infarct size [17]. We found no difference in the peak value of Myo between the two groups of patients, the peak value of CK-MB and cTnI were reduced in the AAβ treated patients. We analyzed that the reasons for no difference in the peak value of Myo between the two groups are as follows: first, Myo has no myocardial specificity, and it is rapidly released into the bloodstream during myocardial infarction, with high sensitivity but poor specificity. Second, Myo increased after 1–4 h of myocardial infarction and reached peak value in 6–7 h; however, some patients see a doctor after 6–7 h of myocardial infarction. The detected peak value of Myo is not the true peak value of Myo during the evolution of myocardial infarction. In any case, the peak value of CK-MB and cTnI were sufficient to represent the myocardial infarct size. The size of the infarct area was indeed reduced in the AAβ treated patients.
In terms of the type of myocardial infarction, the pre-match analysis showed that the proportion of STEMI in the no-AAβ treated patients was higher; therefore, it has more emergency PCI proportion. This finding may be partly explained by the characteristics of the AAβ treated patients because they had a more serious medical history, such as heart failure, stroke and coronary heart disease. As we are known, STEMI was transmural infarction with complete occlusion of the coronary arteries from a pathological point of view, this condition was often worse [18]. The use of AAβ reduces the incidence of STEMI. However, after adjusting for possible confounding variables, the benefits of AAβ disappeared, there was no difference in the type of myocardial infarction between the two groups.
Our present study had limitations inherent to its nonrandomized, observational design. First, similar to previous studies using an administrative database; we did not have full information on the dose and duration of AAβ use. Second, because china population was exclusively included in our study, it is uncertain whether these findings can be applied to other ethnic groups or research institute with different patient characteristics and procedural strategies [19]. Third, the follow-up time was still short, and there was no difference in in-hospital mortality, however, the drugs used in the past really improved the condition of myocardial infarction, which could not negate the long-term effect of the AAβ. Long term follow-up needs to be continued to illustrate the real-world results.
In summary, the use of AAβ prior to myocardial infarction did not improve the in-hospital MACE; this may be the result of a short follow-up. However, AAβ did improve the cardiac function and reduced the infarct size. With the increase in follow-up time, we firmly believe that there must be showing the more benefits of medication and ultimately improving the MACE, These results should be confirmed by future dedicated large, randomized clinical trials with a long term follow-up.
ACEI:
angiotensin-converting-enzyme inhibitors
ARB:
angiotensin-receptor blockers
β-B:
β-blockers
AAβ:
ACEI/ARB/β-B
AMI:
CHD:
Myo:
myoglobin
CK-MB:
creatine kinase-myocardial band
cTnI:
cardiac troponin I
EF:
ejection fraction
FS:
fraction shortening
MACE:
major adverse cardiovascular events
CBD Bank:
Cardiovascular Center Beijing Friendship Hospital Database Bank
STEMI:
ST-segment elevation AMI
NSTEMI:
non-ST-segment elevation AMI
PCI:
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Liu J, Masoudi FA, Spertus JA, et al. Patterns of use of angiotensin-converting enzyme inhibitors, angiotensin receptor blockers among patients with acute myocardial infarction in China from 2001 to 2011: China PEACE-retrospective AMI study. J Am Heart Assoc. 2015. https://doi.org/10.1161/JAHA.114.001343.
Dohi T, Maehara A, Brener SJ, et al. Utility of peak creatine kinase-MB measurements in predicting myocardial infarct size, left ventricular dysfunction, and outcome after first anterior wall acute myocardial infarction (from the INFUSE-AMI trial). Am J Cardiol. 2015;115:563–70.
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LH and CH designed the protocol, DX participated in the clinical data collection, GX drafted the manuscript. All authors read and approved the final manuscript.
We thank Zhao Guoliang for data collection.
The data and materials can be found from the first author and corresponding author.
Consent to publish from the participant to report individual patient data: not applicable (no patient identifier or personalized data shown).
This study was approved by the Institutional Review Board of Beijing Friendship Hospital. All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.
This study was supported by National Natural Science Foundation of China (Grant No. 81600196).
Department of Cardiology, Cardiovascular Center, Beijing Friendship Hospital, Capital Medical University, Beijing, 100050, People's Republic of China
Xuhe Gong, Xiaosong Ding, Hui Chen & Hongwei Li
Department of Internal Medicine, Medical Health Center, Beijing Friendship Hospital, Capital Medical University, Beijing, 100050, People's Republic of China
Hongwei Li
Beijing Key Laboratory of Metabolic Disorder Related Cardiovascular Disease, Beijing, 100069, People's Republic of China
Xuhe Gong
Xiaosong Ding
Correspondence to Hongwei Li.
Gong, X., Ding, X., Chen, H. et al. Real-world use of angiotensin-converting enzyme inhibitors/angiotensin receptor blockers/β-blocks in Chinese patients before acute myocardial infarction occurs: patient characteristics and hospital follow-up. J Transl Med 16, 346 (2018). https://doi.org/10.1186/s12967-018-1720-3
Major adverse cardiovascular events (MACE)


text:
\section{Introduction}
A continuous action  is a triple  $(X, G, \varphi)$  where $X$ is a
compact metric space  with a metric $d$ and  $G$ is  a finitely
generated group with the discrete topology which acts on $X$, such
that the action $\varphi$ is continuous.
  We denote a continuous action $(X, G, \varphi)$ by $\varphi:G\times X\to X$, also we denote by $Act(G;X)$ the set of all
continuous actions $\varphi$ of $G$ on $X$. Let  $S$ be a finite
generating set of $G$.  We consider a metric $d_S$ on $Act(G;X)$ by

\begin{equation*}
d_S(\varphi, \psi)= \sup \{d(\varphi(s, x), \psi(s, x)): x\in X,
s\in S\}
\end{equation*}
for $\varphi, \psi\in Act(G; X)$.\\
A map $f:G\to X$ is called a  $\delta$-pseudo orbit for a continuous
action $\varphi:G\times X\to X$ ( with respect to $S$), if $d(f(sg),
\varphi(s, f(g)))<\delta$ for all $s\in S$ and all $g\in G$. A
$\delta$-pseudo orbit $f:G\to X$ (with respect to $S$ ) is
$\epsilon$-shadowed by $\varphi$-orbit $x\in X$, if $d(f(g),
\varphi(g, x))<\epsilon$ for all $g\in G$.  A continuous action
$\varphi:G\times X\to X$ has shadowing property (with respect to
$S$) if for every $\epsilon>0$ there is a $\delta>0$ such that every
$\delta$-pseudo orbit $f:G\to X$ for $\varphi$ can be $
\epsilon$-shadowed by the $\varphi$-orbit of a point $p\in X$, this
means that $d(f(g), \varphi(g, p))<\epsilon$ for all $g\in G$. The
notion of shadowing property for actions of finitely generated
groups introduced by Osipov and Tikhomirov in \cite{osipov}. They
showed that the shadowing property for actions of finitely generated
groups depends  on both of the  hyperbolic properties of actions of
its elements and  the group structure. For example, if $G$ is a
finitely generated nilpotent group and action of one element in $G$
is hyperbolic, then the group action has the shadowing property
while it cannot be directly generalized to the case of solvable
groups.

 The notion of topological stability for an action
of a finitely generated group on a compact metric space was
introduced by
 Chung
and Lee in  \cite{chung} and they gave a group action version of the
Walter's stability theorem.  Indeed,  a continuous action
$\varphi:G\times X\to X$ is topologically stable (with respect to
$S$) , if for every $\epsilon>0$ there is $\delta>0$ such that for
every continuous action   $\psi:G\times X\to X$  with $d_S(\varphi,
\psi)<\delta$,  there is a continuous map $f:X\to X$ such that
 $d_{C^0}(f, id)<\epsilon$ and $\varphi_gf=f\psi_g$ for all $g\in G$. Moreover, $\varphi$ is called $s$-topologically stable when there exists a surjective  continuous map $f:X\to X$ that satisfies  the mentioned properties.
  If $\varphi:G\times X\to X$ is topologically stable, then  for every $\epsilon>0$ there is $\delta>0$ such that for every $x\in X$ and every continuous action $\psi:G\times X\to X$ with $d_S(\varphi, \psi)<\delta$,  we have $d(\varphi(g, f(x)), \psi(g, x))<\epsilon$ for all $g\in G$. Having this  property is well known to say that the  continuous action $\varphi$ is $\alpha$-persistent. When $\varphi$ is $s$-topologically stable,  for every $\epsilon>0$ there is $\delta>0$ such that for every $x\in X$ and every continuous action $\psi:G\times X\to X$ with $d_S(\varphi, \psi)<\delta$,  we can say that if $f(y)=x$, then  $d(\varphi(g, x ), \psi(g, y))<\epsilon$ for all $g\in G$.
  In this case,  $\varphi$ is called $\beta$-persistent. In other words, a dynamical
system is $\beta$- persistent if its trajectories can be seen on
every small perturbation of it.
Although $s$-topologically stable implies
$\beta$-persistent but topological stability does not imply
$\beta$-persistent. For example, Sakai and Kobayashi \cite{sakai1}
observed that the full shift on two symbols is not
$\beta$-persistent while it is topologically stable. Recently, the
authors in \cite{jung},  introduced a new
tracing property for a homeomorphism $f:X\to X$ referred to as persistent shadowing property and proved that a homeomorphism has persistent shadowing property
 if and only if it has shadowing property and it is $\beta$-persistent. This implies that a homeomorphism has persistent shadowing property if and only if it
  is pointwise persistent shadowable.\\

  \medskip
\noindent

In this paper, we extend   the notion of persistent shadowing
property for a continuous action $\varphi:G\times X\to X$ of some
finitely generated group $G$ on metric space $(X, d)$. Persistent shadowing property is stronger than
of shadowing property and $\beta$-persistent, but in equicontinuous
actions, shadowing and persistent shadowing properties are
equivalent. This implies that every equicontinuous action on the
Cantor space $X$ does have persistent shadowing property. The notion
of persistent shadowing property does  not depend on the choice of a
symmetric finitely generating set and it does not depend on choice
of metric $X$ if $X$ is compact metric space. But Example
\ref{example2}  shows that compactness is essential. Assume that $H$
be a subgroup of $G$. It may be happen that $\varphi:H\times X\to X$
does have the persistent shadowing property
         while $\varphi:G\times X\to X$ does not have it. But in Proposition \ref{syndetic}, we show that if $H$ is a syndetic subgroup of $G$, then the situation
         is different. Also we study relation between persistent shadowing property of $\varphi:G\times X\to X$ and $\varphi_g:X\to X$.
          There is system $\varphi:G\times X\to X$ with persistent shadowing property while $\varphi_g:X\to X$ does not have persistent shadowing property.
         If $G$ is free group, then the situation is different. Indeed in Proposition \ref{op2},
         we show that if $F_2=\langle a, b\rangle$ is a free group and $\varphi:F_2\times X\to X$ has shadowing property, then $\varphi_{a^{-1}b}:X\to X$ has
         persistent shadowing property. Also, one can check that these results do hold for notions of shadowing property, $\alpha$-persistent and $\beta$-persistent, see Remark\ref{224} and Remark \ref{225}\\
\medskip
\noindent

 Recently, in \cite{ali2}, we introduced the notion of
compatibility of a measure with respect to $\alpha$-persistent. We extend this notion with respect persistent shadowing property and the set of
compatibility measures with persistent shadowing property for
$\varphi$ is denoted by $\mathcal{M}_{PSh}(X, \varphi)$, see  Subsection \ref{s22}. We show that
$\mathcal{M}_{PSh}(X, \varphi)$ is an $F_{\sigma\delta}$ subset of
$\mathcal{M}(X)$ and for $\mu\in\mathcal{M}_{PSh}(X, \varphi)$ and
homeomorphism $f:X\to Y$, we have  $f_*(\mu)\in
\mathcal{M}_{PSh}(Y, f\circ \varphi\circ f^{-1})$  where $f\circ
\varphi\circ f^{-1}:G\times Y\to Y$ is defined by
        $f\circ \varphi\circ f^{-1}(g, x)=f\circ \varphi_g\circ f^{-1}(x)$, see Proposition \ref{kj}. In Proposition \ref{pki}, we show that
         if measure $\mu$ is compatible with persistent shadowing property  for continuous action $\varphi$, then $\varphi$ does have persistent shadowing property on
          $supp(\mu)$. This implies that if every non-atomic probability measure is compatible with persistent shadowing property  for continuous action $\varphi$,
         then $\varphi$ has persistent shadowing property. Also, we introduce compatibility a measure with respect shadowing property, $\alpha$-persistent and
         $\beta$-persistent for continuous action $\varphi:G\times X\to X$ and denote them by
         $\mathcal{M}_{Sh}(X, \varphi)$, $\mathcal{M}_\alpha(X, \varphi)$ and $\mathcal{M}_{\beta}(X, \varphi)$, respectively.  Results of Proposition \ref{pki}
          can be obtain for compatibility a measure in the case of shadowing property and
$\beta$-persistent, see Remark \ref{pkii}.

         In Section 3, we introduce the notions of   persistent shadowable points, uniformly $\alpha$-persistent point, uniformly $\beta$-persistent point
         and denote them by $PSh(\varphi)$, $UPersis_\alpha(\varphi)$ and $UPersis_\beta(\varphi)$, respectively. Also, we recall  the notions of shadowable points,
          $\alpha$-persistent points, $\beta$-persistent points for continuous action
$\varphi:G\times X\to X$ and denote them by $Sh(\varphi)$,
$Persis_\alpha(\varphi)$ and $Persis_\beta(\varphi)$, respectively.
Although $Sh(\varphi)\subseteq UPersis_\alpha(\varphi)\subseteq
Persis_\alpha(\varphi)$ but Example \ref{non-shadowable} shows that
the $Sh(\varphi)\neq UPersis_\alpha(\varphi)$ and
$Upersis_\alpha(\varphi)\neq Persis_\alpha(\varphi)$. For
equicontinuous action $\varphi:G\times X\to X$, we have
$UPersis_\beta(\varphi)=Persis_\beta(\varphi)$ and
$Persis_\alpha(\varphi)\subseteq Persis_\beta(\varphi)$. Moreover,
if $X$ is generalized homogeneous compact metric space, then
$Sh(\varphi)=UPersis_\alpha(\varphi)=Persis_\alpha(\varphi)$, see Proposition \ref{u}.\\
In Subsection \ref{400}, we study persistent shadowable point for a
group action and in item 3 of Proposition \ref{wok}, we show that
   Continuous action $\varphi:G\times X\to X$  has the persistent shadowing property if and only if it is pointwise persistent shadowable. Also in item 4 of
    Proposition \ref{wok}, we prove that $PSh(\varphi)= UPersis_\beta(\varphi)\cap Sh(\varphi)$. This implies that continuous action $\varphi:G\times X\to X$ has
     persistent shadowing property if and only if it is $\beta$-persistent and it has shadowing
     property.\\
    In Subsection \ref{401}, we study  various shadowable points via
    measure theory. Indeed,  for continuous action $\varphi:G\times X\to X$ on compact metric space $(X,
    d)$  and Borel probability measure
    $\mu$,   we show that
 $\mu\in M_{PSh}(X, \varphi)\Leftrightarrow
    supp(\mu)\subseteq PSh(\varphi)$, see Proposition \ref{Lb}. In Proposition \ref{Lbb} we show that
$\mu(\overline{PSh(\varphi))}=1\Leftrightarrow
\mu\in\overline{\mathcal{M}_{PSh}(X,
    \varphi)}$ and in Proposition \ref{12354}, we show that $\overline{PSh(\varphi)}=PSh(\varphi)$ if and only if
$\overline{\mathcal{M}_{PSh}(X, \varphi)}=\mathcal{M}_{PSh}(X,
\varphi)$. Note that, result of this paragraph, obtain for other
types of shadowing property, see Proposition \ref{12355}. In
equicontinuous action $\varphi:G\times X\to X$,
$Persis_\beta(\varphi)=UPersis_\beta(\varphi)$ is closed set in $X$,
hence we can say that $\overline{\mathcal{M}_{\beta}(X,
\varphi)}=\mathcal{M}_\beta(X, \varphi)$ if $\varphi:G\times X\to X$
is equicontinuous action. This implies that
$\mu(Persis_\beta(\varphi))=1$ if and only if
$\mu\in\mathcal{M}_\beta(X, \varphi)$, whenever $\varphi:G\times
X\to X$ is equicontinuous action.
 Finally, In Proposition  \ref{3214}, we show that
$\overline{\mathcal{M}_{PSh}(X, \varphi)}=\mathcal{M}(X)
    \Leftrightarrow \overline{PSh(\varphi)}=X$, $\overline{\mathcal{M}_{Sh}(X, \varphi)}=\mathcal{M}(X)
    \Leftrightarrow \overline{Sh(\varphi)}=X$,  $\overline{\mathcal{M}_\beta(X, \varphi)}=\mathcal{M}(X)
    \Leftrightarrow \overline{Persis_\beta(\varphi)}=X$ and  $\overline{\mathcal{M}_{\alpha}(X, \varphi)}=\mathcal{M}(X)
    \Leftrightarrow \overline{Persis_\alpha(\varphi)}=X$.

\section{Persistent shadowing property }
In this section, firstly, we extend  the notion of persistent shadowing property  from \cite{jung} to group actions and study it.
     \begin{definition}\label{def21}
 A continuous action $\varphi:G\times X\to X$ has persistent shadowing property ( with respect $S$) if for every $\epsilon>0$ there is $\delta>0$ such that
  every $\delta$-pseudo orbit $f:G\to X$ for $\psi:G\times X\to X$ with $d_S(\varphi, \psi)<\delta$ can be $(\psi, \epsilon)$-shadowed by a point $p\in X$.

\end{definition}
It is not hard to see that  notion of persistent shadowing property
does  not depend on the choice of a symmetric finitely generating
set. Also one can check that the this notion does not depend on
choice of metric $X$ if $X$ is compact metric space. The following
example shows that compactness is essential.

\begin{example}\label{example2}
Let $T:\mathbb{R}\to S^1\setminus\{(0, 1)\}$ be a map given by
\begin{equation*}
T(t)=(\frac{2t}{1+t^2}, \frac{t^2-1}{t^2+1}), \quad \text{for all
}t\in\mathbb{R},
\end{equation*}
and let $X=T(\mathbb{Z})$. Let $d'$ be the metric on $X$ induced by
the Riemannian metric on $S^1$, and let $d$ be a discrete metric on
$X$. It is clear that $d$ and $d'$ induce the same topology on $X$.
Let $g_1 :X\to X$ be a homeomorphism defined by $g(a_i)=a_{i+1}$ and
$g_2:X\to X$ be defined by $g_2(a_i)= a_{i+2}$. Consider  action
$\varphi:G\times X\to X$ generated $g_1, g_2:X\to X$. Since the
metric $d$ is discrete, one can see that $\varphi$ does have
persistent shadowing property. By contradiction, let $\varphi$ has
persistent shadowing property with respect to $ d' $. Hence it does
have shadowing property with respect to $d'$.

 For $\epsilon=\frac{1}{2}$ and $z\in X$, let $\delta>0$ be
an $ \epsilon $-modulus of  shadowing property of  continuous action
$\varphi$. Choose $k\in\mathbb{N}$ satisfying $d'(a_k,
a_{-k})<\frac{\delta}{2}$, and consider a homeomorphisms $f_i:X\to
X$, $i=1, 2$, given by
 \[ f_1(a_i)= \left\lbrace
  \begin{array}{c l}
     a_{i+1}, & \text{\rm{ $ i\in\{-k, \ldots, k-1\}$}},\\
a_{-k}, & \text{\rm{$i=k$}},\\
a_i, & \text{\rm{otherwise}}.
  \end{array}
\right. \] and
 \[ f_2(a_i)= \left\lbrace
  \begin{array}{c l}
     a_{i+2}, & \text{\rm{ $ i\in\{-k, \ldots, k-2\}$}},\\
     a_{-k+1}, &\text{\rm{$i=k-1$}}\\
a_{-k}, & \text{\rm{$i=k$}},\\
a_i, & \text{\rm{otherwise}}.
  \end{array}
\right. \]

 Since $d'(f_i(x), g_i (x))<\delta$ for all
$x\in X$, hence if $\psi:G\times X\to X$ generated by $f_1, f_2$,
then $d(\varphi, \psi)<\delta$ and for $x\in X$   there is $z\in X$
such that $d'(\varphi(g, z), \psi(g, x))<\epsilon$, this implies
that  $d'(g_{i}^n(z), f^n_i(y))<\epsilon$. Since $\{g_1^n(z), n\in
\mathbb{Z}\}=X$, so we can find an integer $ n\in \mathbb{Z}$ such
that $d'(g^n_1(z),f^n_1(y))\geq\epsilon$, which is a contradiction.
Therefore $\varphi$ does have persistent shadowing property with
respect to $ d $ but it does not have  persistent shadowing property
with respect to $ d' $.
\end{example}

\subsection{The Action of Syndetic Subgroups}\label{s20}

Let $BS(1, n)=\langle a, b: ba = a^nb \rangle$ and $\varphi:G\times
\mathbb{R}^2\to \mathbb{R}^2$ be generated by $f_a(x)=Ax$ and
$f_b(x)=Bx$ where
\begin{equation}
A=\left
\begin{array}{cc}
  1 & 0 \\
  1 & 1 \\
\end{array
\right)  \;\& \;\;
B=\left
\begin{array}{cc}
  \lambda & 0 \\
  0 & n\lambda \\
\end{array
\right)
\end{equation}
Then for $1<\lambda\leq n$ and $n>1$, $f_b$ has persistent shadowing
property. In \cite{jung}, it is shown that the homeomorphism $f:X\to
X$ does have persistent shadowing property if and only if it does
have shadowing property and it is $\beta$-persistent. Hence if
$H=\langle b\rangle$ is a subgroup of $B(1, n)$, then
$\varphi|H:H\times \mathbb{R}^2\to \mathbb{R}^2$ has persistent
shadowing property while by \cite[Theorem 4.4(1)]{osipov},
$\varphi:G\times \mathbb{R}^2\to \mathbb{R}^2$ does not have
shadowing property.
In this subsection, we show that if $H$ is a syndetic subgroup of $G$, then situation is different.\\
 Let $||g||_{S}$ denote the length of the shortest representation  of the element $g$ in term of element from $S$. For continuous action $\varphi:G\times X\to X$   on compact metric space $(X,d)$, $\epsilon>0$ and $k\in\mathbb{N}$, there is $\delta>0$ such that for all $g\in G$ with $||g||_S<k$
\begin{align}\label{u}
d(x, y)<\delta\Rightarrow d(\varphi(g, x), \varphi(g,
y))<\frac{\epsilon}{k}
\end{align}
By triangle inequality, it can to see that the following lemma is
true.
\begin{lemma}\label{pseudo}
Let $S$ be a finitely generating set of $G$ and $\varphi:G\times
X\to X$ be a continuous action on compact metric space $(X,d)$. For
$\epsilon>0$ and $N\in\mathbb{N}$, there is $\delta>0$ such that if
$f:G\to X$ is a $\delta$-pseudo orbit, then for every $h\in G$ with
$||h||_S<N$ and every $g\in G$ we have $d(f(hg), \varphi(h,
f(g)))<\epsilon$
\end{lemma}
Subset $H\subseteq G$ is syndetic if there is finite set $F\subseteq
G$ such that $G=FH$. Hence a subgroup $H$ is syndetic in $G$ if and
only if it is finite index subgroup of $G$ i.e. there is finite set
$\{g_i\}_{i=1}^n$ such that $G= \bigcup_{i=1}^n g_iH$.
\begin{proposition}\label{syndetic}
Let $H$ be a finite index subgroup of $G$. Then continuous action
$\varphi:G\times X\to X$ has persistent shadowing property if
$\varphi:H\times X\to X$ has persistent shadowing property.
\end{proposition}
\begin{proof}
Let $H$ be finite index subgroup of $G$. Let $A$ be a symmetric
finitely generating set of $H$. We can add more elements to $A$ to
get a symmetric finitely generating set $S$ of $G$.  Also let $G=
\bigcup_{i=1}^n g_iH$ and $N=\max \{ ||g_i||_S: 1\leq i \leq n\}$.

 Let $\epsilon>0$ be given. Choose $\delta>0$ such that  for all $g\in G$ with $||g||_S<N$
\begin{align}\label{k}
d(x, y)<\delta\Rightarrow d(\varphi(g, x), \varphi(g,
y))<\frac{\epsilon}{N}
\end{align}
Choose $\eta>0$ corresponding to $\delta>0$ and $N\in\mathbb{N}$
that satisfies Lemma\ref{pseudo}.\\
 Let $\epsilon>0$ be given. By triangle inequality, it is easy to see that for $\epsilon>0$ and $N$ above, there is $\eta>0$
 such that if $d_S(\varphi, \psi)<\eta$, and $d(a, b)<\eta$, then

 \begin{equation}\label{b}
     d(\varphi(g, a), \psi(g, b))<\frac{\epsilon}{N},  \forall g\in G \text{ with } ||g||_{S} < N.
 \end{equation}
 and
 \begin{equation}\label{pl}
d(\psi(g, a), \psi(g, b))<\frac{\epsilon}{N}, \text{ for all }
||g||_S<N.
 \end{equation}
Let $\varphi:H\times X\to X$ has persistent shadowing property, we
show that $\varphi:G\times X\to X$ has persistent shadowing
property. Let $\epsilon>0$ be given. Choose $\delta>0$ that
satisfies Lemma \ref{pseudo} and Relation \ref{b}, Relation
\ref{pl}. Choose $\eta>0$ corresponding $\frac{\delta}{2}>0$  by
definition of persistent shadowing property of $\varphi|H$. We show
that every $\eta$-pseudo orbit $F:G\to X$ for continuous action
$\psi:G\times X\to X$ with $d_S(\varphi, \psi)<\eta$ can
$\psi$-shadowed by a point of $X$. The map $F:G\to X$ is
$2\eta$-pseudo orbit for $\varphi:G\times X\to X$, hence
    \begin{equation}\label{kl}
      d(F(gh), \varphi(g, F(h)))<\frac{\epsilon}{N}, \text{  for all } g \in G \text{ with } ||g||_S<N.
    \end{equation}
    Since $F:H\to X$ is $\eta$-pseudo orbit for $\psi:H\times X\to X$ with $d_A(\varphi, \psi)<\eta$,
    hence by persistent shadowing property of $\varphi|H$ there is $p\in X$ such that $d(F(h), \psi(h, p))<\frac{\delta}{2}$.
     Also by Relation \ref{pl} we have $d(\varphi(g_i, F(h)), \psi(g_i, \psi(h, p)))<\frac{\epsilon}{N}$. Hence by Relation \ref{kl}, we have
      $d(F(g_ih), \psi(g_ih, p))<\epsilon$ i.e. $d(F(g), \psi(g, p))<\epsilon$ for all $g\in G$.

\end{proof}
\begin{remark}\label{224}
Let $P$ be one of the following property:  $(a)$ shadowing property,
$(b)$ $\alpha$-persistent, $(c)$ $\beta$-persistent. Similar to
proof of Proposition \ref{syndetic}, if $H\leq G$ is syndetic subset
of $G$ and continuous action $\varphi:H\times X\to X$ does have $P$-
property, then $\varphi:G\times X\to X$ has $P$-property.
\end{remark}
\subsection{ The Action Of Free Groups}\label{s21}
Group  $BS(1, n)$ is solvable group, hence it is not free group. In
the case of finitely free group actions $G$ , by \cite[Theorem
4.9.]{osipov}, if $\varphi:G\times X\to X$ has shadowing property,
then $\varphi_g:X\to X$ has shadowing property, for all $g\in G$. In
the following, we extend it in the case of  persistent shadowing
property.
\begin{proposition}\label{op2}
Let $\varphi:G\times X\to X$ be a continuous action of free group
$F_2=\langle a, b\rangle$ on compact metric space $(X, d)$.
 If $\varphi:F_2\times X\to X$ has persistent shadowing property, then $\varphi_{a^{-1}b}:X\to X$ has persistent shadowing property.

\end{proposition}
\begin{proof}
  Let $\epsilon>0$ be given. Choose $\epsilon_0>0$ corresponding to $\epsilon>0$ by persistent shadowing property of $\varphi:G\times X\to X$. For $\epsilon_0>0$ there is $\delta>0$ such that for every continuous action $\psi:F_2\times X\to X$ that is $\delta$-close to $\varphi:G\times X\to X$, we have
        \begin{equation*}
          d(x, y)<\delta\Rightarrow d(\psi(g, x), \psi(g, y))<\epsilon, |g|_S\leq 2,
        \end{equation*}
        Let $\{x_n\}_{n\in\mathbb{Z}}$ be $\delta$ pseudo orbit of homeomorphism $f:X\to X$ wit $d(f, \varphi_{a^{-1}}\varphi_b)<\delta$.
         It is easy to see that if $\psi:F_2\times X\to X$ is generated by $\psi_a=\varphi_a$ and $\psi_b= \varphi_a\circ f$, then $\psi:F_2\times X\to X$
         is $\epsilon_0$- close to $\varphi:F_2\times X\to X$. Define $K: F_2\to X$ define by $K(t)= \psi(v, x_k)$ where $v\in G$ is an element with minimal
         length such that $t=v(a^{-1}b)^k$ for some $k\in \mathbb{Z}$. It is not hard to see that $K:G\to X$ is $\epsilon_0$- pseudo orbit of continuous
          action $\psi:F_2\times X\to X$ with $d(\varphi, \psi)<\epsilon_0$. By persistent shadowing property, there is $y\in X$ such that
          $d(K(g), \psi(g, y))<\epsilon$ for all $g\in G$. Since $K((a^{-1}b)^k)= x_k$ and $\psi((a^{-1}b)^k, y)= f^k(y)$, we have $d(f^k(y), x_k)<\epsilon$ for
          all $k\in \mathbb{Z}$.

\end{proof}

\begin{remark}\label{225}
Let $P$ be one of the following property:  $(a)$ shadowing property,
$(b)$ $\alpha$-persistent, $(c)$ $\beta$-persistent. Similar to
proof of Proposition \ref{op2}, we can show that if continuous
action $\varphi:F_2\times X\to X$ of free group $F_2=\langle a,
b\rangle$ on compact metric space $(X, d)$ has $P$- property, then
$\varphi_{a^{-1}b}:X\to X$ has $P$- property.
\end{remark}
\subsection{Persistent shadowing property and related
measure}\label{s22}
 For continuous action $\varphi:G\times X\to X$,
$\epsilon>0$, $\delta>0$ and generating set $S$, we denote
$PSh_{\varphi, S}(\delta, \epsilon)$  the set of all $x\in X$ such
that every $\delta$-pseudo orbit $f:G\to X$ of continuous action
$\psi$ with $d_S(\varphi, \psi)<\delta$ and $f(e)=x$, respect to
generating set $S$, can be $(\epsilon, \psi)$-shadowed by a point in
$X$.

It is clear that
\begin{enumerate}
\item Let $S, S'$ be generating sets for $G$ and $\epsilon>0$ be given. Then for every $\delta>0$ there is $\eta>0$ such that $PSh_{\varphi, S}(\eta, \epsilon)\subseteq PSh_{\varphi, S'}(\delta, \epsilon)$
  \item Continuous action $\varphi:G\times X\to X$ does have persistent shadowing property with respect to generating set $S$ if and only if for every $\epsilon>0$ there is $\delta>0$ such that $PSh_{\varphi, S}(\delta, \epsilon)=X$.
      \item If continuous action $\varphi:G\times X\to X$ does have persistent shadowing property on compact set $K\subseteq X$, then for every $\epsilon>0$ there exist a neighborhood $U$ of $K$ and  $\delta>0$ such that $U\subseteq PSh_{\varphi, S}(\delta, \epsilon)$.
          \item $PSh_{\varphi, S}(\delta, \epsilon)$ is a closed subset of $X$.
\end{enumerate}
The Borel $\sigma$-algebra of $X$ is the $\sigma$-algebra
$\mathcal{B}(X)$ generated by the open subsets of $X$.
 A Borel probability measure is a $\sigma$-additive measure $\mu$ defined in $\mathcal{B}(X)$ such that $\mu(X)=1$.
 We denote by $\mathcal{M}(X)$ the set of all Borel probability measures of $X$. This set is convex and  compact metrizable if it endowed with weak$^*$ topology:
 the one ruled by the convergence $\mu_n\to \mu$ if and only if $\int f d\mu_n\to \int f d\mu$ for every continuous map $f:X\to \mathbb{R}$.\\
\begin{definition}\label{plr}
 A measure $\mu\in\mathcal{M}(X)$ is compatible with the persistent
shadowing property  for the continuous action $\varphi:G\times X\to
X$, $\mu\in \mathcal{M}_{PSh}(X, \varphi)$ if for every $\epsilon>0$
there is $\delta>0$ such that if $\mu(A)>0$, then
\begin{equation*}
A\cap Sh_{\psi}(\delta, \epsilon)\neq \emptyset.
\end{equation*}
for every continuous action $\psi:G\times X\to X$ with
$d_S(\varphi,\psi)<\delta$.

\end{definition}

\begin{example}\label{exam1}
Let  continuous action $\varphi:G\times X\to X$ admits  an
$\varphi$-invariant measure and let $\varphi$  does have  persistent
shadowing property on the non-wandering set $\Omega(\varphi)$. Then
every $\varphi$-invariant Borel probability measure $\mu$ on $X$ is
compatible with the property of persistent shadowing property.
 \end{example}

 Let $\mu\in\mathcal{M}_{PSh}(X, \varphi)$ , $\epsilon>0$ and $h:(X, d)\to (Y, \rho)$ be a homeomorphism. We will show that there is $\delta>0$ such that
\begin{equation}
h_{*}(\mu)(B)>0 \Rightarrow B\cap PSh_{h\circ \varphi \circ
h^{-1}}(\delta, \epsilon)\neq \emptyset.
\end{equation}
For $\epsilon>0$ there is $\epsilon_0>0$ such that
\begin{equation}\label{rty}
d(a, b)<\epsilon_0\Rightarrow \rho(h(a), h(b))<\epsilon.
\end{equation}
For $\epsilon_0>0$ there is $\epsilon_1>0$ in definition of
$\mu\in\mathcal{M}_{PSh}(X, \varphi)$. Since $\mu(h^{-1}(B))>0$,
hence $h^{-1}(B)\cap \mathcal{M}_{PSh}(\epsilon_1, \epsilon_0)\neq
\emptyset$. For $\epsilon_1>0$ there is $\delta>0$ such that
\begin{equation}
\rho(c, d)<\delta\Rightarrow d(h^{-1}(c), h^{-1}(d))<\epsilon_1.
\end{equation}
 Fix $x\in h^{-1}(B)\cap PSh_\varphi(\epsilon_1, \epsilon_0)$. One can check that if
 $F':G\to Y$ be $\delta$-pseudo orbit of continuous action
 $\psi':G\times Y\to Y$ with $\rho_S(f\circ \varphi\circ f^{-1},
 \psi')<\delta$ and $F'(e)=h(x)$, then $F:G\to X$ defined by
 $F(g)=h^{-1}(F'(g))$ is $\epsilon_1$-pseudo orbit of continuous
 action $h^{-1}\circ \psi'\circ h $ with $d_S(\varphi,
 h^{-1}\circ \psi'\circ h)<\epsilon_1$ and $F(e)=x$. By $x\in h^{-1}(B)\cap PSh_\varphi(\epsilon_1,
 \epsilon_0)$, there is $p\in X$ such that $d(F(g), h^{-1}\circ
 \psi'_g\circ h(p))<\epsilon$ for all $g\in G$. By relation
 \ref{rty}, we have $d(h\circ F(g), \psi'_g(h(p)))<\epsilon$ i.e.
 $d(F'(g), \psi'_g(h(p)))<\epsilon$. This means that $h(x)\in B\cap
 PSh_{h\circ \varphi \circ h^{-1}}(\delta, \epsilon)$.
\begin{itemize}
\item  If $h:(X, d)\to(Y, \rho)$ is a homeomorphism and $\mu\in  \mathcal{M}_{PSh}(X, \varphi)$, then $h_{*}(\mu)\in \mathcal{M}_{PSh}(Y, h\circ \varphi\circ h^{-1})$,
\item If $G$ is an abelian group, then $\mathcal{M}_{PSh}(X, \varphi)$ is $(\varphi_g)_*$-invariant, for all $g\in G$.
\end{itemize}
It is known that every continuous action of a countable abelian group on compact metric space  admit  an $\varphi$-invariant measure. Also, One can check that  $\mathcal{M}_{PSh}(X, \varphi)$ is a convex  subset of $\mathcal{M}(X)$, hence by Proposition 2.12
in \cite{ali2}, the following item does hold.
\begin{itemize}
\item If $G$ is  an abelian group, then $\overline{\mathcal{M}_{PSh}(X, \varphi)}$ contains
a $\varphi$-invariant measure.
\end{itemize}
It is known that a  group action $\varphi:G\times X\to X$ on the
compact metric space $X$ has a $\varphi$- invariant Borel
probability measure on $X$ if and only if $G$ is amenable, see
\cite{1}. Hence, the following group action where  $G=SL(2, \mathbb{Z})$ and $X=\mathbb{R}\cup\{\infty\}$ does not contan any invarant meausre.\\

\begin{equation*}
  \varphi
  (\left(
  \begin{array}{ll}
  a & b\\
  c & d
  \end{array}
  \right),
  z)=\frac{az+b}{cz+d}.
\end{equation*}

 Take $$\mathcal{C}_{PSh(\varphi)}(\delta,
 \epsilon)=\{\mu\in\mathcal{M}(X): \mu(PSh_\varphi(\delta,
 \epsilon))=1\}.$$ It is easy to see that $\mathcal{C}_{PSh(\varphi)}(\delta,
 \epsilon)$ is a convex and closed subset of $\mathcal{M}(X)$ for
 every $\epsilon>0$ and any $\delta>0$.
 \begin{proposition}\label{kj}
 Let $\varphi:G\times X\to X$ be a continuous action. Then

 \begin{enumerate}
    \item $ \mathcal{M}_{PSh}(X, \varphi)=\bigcap_{n\in\mathbb{N}}(\bigcup_{m\in\mathbb{N}}(\bigcap_{l\in\mathbb{N}}C_{PSh(\varphi)}(n^{-1}+l^{-1}, m^{-1})))$

    \item The subset $\mathcal{M}_{PSh}(X, \varphi)$ is an
    $F_{\sigma\delta}$ subset of $\mathcal{M}(X)$.
 \end{enumerate}
 \end{proposition}
 \begin{proof}
 \begin{enumerate}
 \item Fix a $\mu\in\mathcal{M}_{PSh}(X, \varphi)$ and an $n\in\mathbb{N}$.
Choose a $\delta>0$ such that if $\mu(A)>0$ then $A\cap
PSh_\varphi(\delta, \frac{1}{n})\neq \emptyset$.Choose
$m\in\mathbb{N}$ such that $m^{-1}<\delta$. Note that if $A\cap
PSh_\varphi(\delta, \frac{1}{n})\neq \emptyset$, then $A\cap
PSh_\varphi(\frac{1}{m}, \frac{1}{n})\neq \emptyset$. This implies
that
$\mu\in\bigcup_{m\in\mathbb{N}}(\bigcap_{l\in\mathbb{N}}C_{PSh(\varphi)}(
m^{-1}, n^{-1}+l^{-1})))$. Conversely choose a
$\mu\in\bigcap_{n\in\mathbb{N}}(\bigcup_{m\in\mathbb{N}}(\bigcap_{l\in\mathbb{N}}C_{PSh(\varphi)}(
m^{-1}, n^{-1}+l^{-1})))$. Thus, for every $n\in\mathbb{N}$, there
is $k\in\mathbb{N}$ such that $\mu\in
\bigcap_{l\in\mathbb{N}}C_{PSh(\varphi)}(\frac{1}{k}, \frac{1}{n}+
\frac{1}{l})$. This means that for every $n\in\mathbb{N}$ there is
$k\in\mathbb{N}$ such that $\mu\in
\bigcap_{l\in\mathbb{N}}C_{PSh(\varphi)}(\frac{1}{k}, \frac{1}{n}+
\frac{1}{l})$. This implies that for every $\epsilon>0$ there exist
$N, K\in\mathbb{N}$ such that $(\frac{1}{N}+\frac{1}{K})<\epsilon$
and $\mu\in C_{PSh(\varphi)}(\frac{1}{N}+\frac{1}{K}, \frac{1}{K})$.
Therefor, for every $\epsilon>0$ choose $\delta=\frac{1}{K}$ to
conclude then $\mu\in\mathcal{M}_{PSh(\varphi)}(X)$.
\item Since $C_{PSh(\varphi)}(
m^{-1}, n^{-1}+l^{-1})))$ is a closed subset of $\mathcal{M}(X)$ and
a countable intersection of closed sets is closed, hence we can say
that  $(\bigcap_{l\in\mathbb{N}}C_{PSh(\varphi)}( m^{-1},
n^{-1}+l^{-1})))$ is a closed subset of $\mathcal{M}(X)$ for every
pair $m, n\in\mathbb{N}$. Therefor
$\bigcup_{m\in\mathbb{N}}(\bigcap_{l\in\mathbb{N}}C_{PSh(\varphi)}(
m^{-1}, n^{-1}+l^{-1})))$ is an $F_{\sigma\delta}$ subset of
$\mathcal{M}(X)$, for every $n\in\mathbb{N}$ and hence by item (1),
$\mathcal{M}_{PSh}(X,\varphi)$ is an $F_{\sigma\delta}$  subset of
$\mathcal{M}(X)$.
\end{enumerate}
 \end{proof}

\begin{proposition}\label{pki}
  Let $\varphi:G\times X\to X$ be a continuous action on compact metric space $(X, d)$ and $\mu\in\mathcal{M}_{PSh}(\varphi)$.
  Then $\varphi$ has persistent shadowing property on $supp(\mu)$.
\end{proposition}
\begin{proof}
 Let $\epsilon>0$ be given. Choose $0<\delta<\frac\epsilon2$ such that for every continuous action $\psi:G\times X\to X$ with $d_S(\varphi, \psi)<\delta$ we have
  \begin{equation*}
  \mu(A)>0\Rightarrow A\cap Sh_{\delta, \frac\epsilon2}(\psi)\neq \emptyset
\end{equation*}

For $\delta>0$ there is $0<\eta<\frac\delta2$ such that for every
continuous action $\psi:G\times X\to X$ with $d_S(\varphi,
\psi)<\eta$ we have
\begin{equation*}
  d(a, b)<\eta\Rightarrow d(\psi_s(a), \psi_s(b))<\frac\delta2, \forall s\in S,
\end{equation*}
We claim that if $d_S(\varphi, \psi)<\delta$ and $F:G\to X$ is an  $\eta$-pseudo orbit for   $\psi:G\times X\to X$ with  $F(e)=p\in supp(\mu)$, then there is $z\in X$ such that $d(F(g), \psi(g, z))<\epsilon$ for all $g\in G$. \\

By $p\in supp(\mu)$ we have $\mu(B_\eta(p))>0$, this implies that
$B_\eta(p)\cap S_{\delta, \epsilon}(\psi)\neq \emptyset$ for
$\psi:G\times X\to X$. Take $q\in B_\eta(p)\cap Sh_{\delta,
\epsilon}(\psi)$ and  define $f:G\to X$ by $f(e)=q$ and $f(g)=F(g)$
for all $g\neq e$. It is easy to see that $f:G\to X$ is a
$\delta$-pseudo orbit of $\psi$. By $f(e)\in Sh_{\delta,
\epsilon}(\psi)$, there is $y\in X$ with $d(f(g), \varphi(g,
y))<\frac\epsilon2$ for all $g\in G$. This implies that $d(F(g),
\psi(g, y))<\epsilon$ for all $g\in G$.

\end{proof}

It is known that the set of Borel probability measures of a compact
metric space $X$ with support equals to $X$ is a dense $G_\delta$
subset of $\mathcal{M}(X)$, see \cite[Lemma 3.6]{baut}, also if $X$
has no isolated point,  then the non-atomic Borel probability
measures is a dense $G_\delta$ subset of $\mathcal{M}(X)$ see
\cite[Corollary 8.2]{par}, thus since $X$ is a compact space, we can
say that if $X$ is a compact space without isolated point, then the
set of non-atomic Borel probability measures with support equals to
$X$ is dense in $\mathcal{M}(X)$.  Hence by Proposition \ref{pki},
we have
\begin{corollary}
  Let $\varphi:G\times X\to X$ be a continuous action of a compact metric space $X$ without
isolated point.If every non-atomic Borel probability measure $\mu$
is compatible with persistent shadowing property for
$\varphi:G\times X\to X$, then $\varphi$ has persistent shadowing
property.
\end{corollary}
For continuous actions $\varphi, \psi:G\times X\to X$,  and $x\in
X$, we denote

\begin{equation*}
\Gamma_\epsilon^{\varphi,\psi}(x)= \bigcap_{g\in G}\varphi (g^{-1},
B[\psi(g, x), \epsilon])=\{y\in X: d(\varphi(g, y), \psi(g, x))\leq
\epsilon \text{ for every } g\in G\}
\end{equation*}
and
\begin{equation*}
B(\epsilon, \varphi, \psi)=\{x\in X: \Gamma_\epsilon^{\varphi,
\psi}(x)\neq \emptyset\}.
\end{equation*}
It is easy to see that $B(\epsilon, \varphi, \psi)$ is a compact set
in $X$.

We say that
\begin{enumerate}
\item A measure $\mu\in\mathcal{M}(X)$ is compatible with the
shadowing property  for the continuous action $\varphi:G\times X\to
X$, $\mu\in \mathcal{M}_{Sh}(X, \varphi)$ if for every $\epsilon>0$
there is $\delta>0$ such that if $\mu(A)>0$, then
\begin{equation*}
A\cap Sh_{\varphi}(\delta, \epsilon)\neq \emptyset.
\end{equation*}
\item (\cite{ali2}) A measure $\mu\in\mathcal{M}(X)$ is compatible with the
$\alpha$-persistent  for the continuous action $\varphi:G\times X\to
X$, $\mu\in \mathcal{M}_\alpha(X, \varphi)$, if for every
$\epsilon>0$ there is $\delta>0$ such that if $\mu(A)>0$, then
\begin{equation*}
A\cap B(\epsilon, \varphi, \psi)\neq \emptyset.
\end{equation*}
for every continuous action $\psi:G\times X\to X$ with
$d_S(\varphi,\psi)<\delta$.
  \item A measure $\mu\in\mathcal{M}(X)$ is compatible with the
$\beta$-persistent  for the continuous action $\varphi:G\times X\to
X$, $\mu\in\mathcal{M}_\beta(X, \varphi)$, if for every $\epsilon>0$
there is $\delta>0$ such that if $\mu(A)>0$, then
\begin{equation*}
A\cap B(\epsilon, \psi, \varphi)\neq \emptyset.
\end{equation*}
for every continuous action $\psi:G\times X\to X$ with
$d_S(\varphi,\psi)<\delta$.
\end{enumerate}

\begin{remark}\label{pkii}
With similar proof of Proposition \ref{kj},  one can check that
$\mathcal{M}_{Sh}(X, \varphi)$, $\mathcal{M}_\alpha(X, \varphi)$ and
$\mathcal{M}_\beta(X, \varphi)$ are $F_{\sigma\delta}$ subsets of
$\mathcal{M}(X)$. Also  for homeomorphism $h:(X, d)\to (Y,\rho)$, if
$\mu\in \mathcal{M}_{Sh}(X, \varphi)$, $\mu\in
\mathcal{M}_{\alpha}(X, \varphi)$ and $\mu\in \mathcal{M}_{\beta}(X,
\varphi)$, then $f_{*}(\mu)\in\mathcal{M}_{Sh}(Y, h\circ
\varphi\circ h^{-1})$, $f_{*}(\mu)\in\mathcal{M}_\alpha(Y, h\circ
\varphi\circ h^{-1})$, $f_{*}(\mu)\in\mathcal{M}_\beta(Y, h\circ
\varphi\circ h^{-1})$, respectively. With similar technics in
Proposition \ref{pki}, we can show that if $\mu\in
\mathcal{M}_{Sh}(X, \varphi)$, $\mu\in \mathcal{M}_{\alpha}(X,
\varphi)$ and $\mu\in \mathcal{M}_{\beta}(X, \varphi)$, then
$\varphi:G\times X\to X$ does have shadowing property,
$\alpha$-persistent and $\beta$-persistent on $supp(\mu)$,
respectively.
\end{remark}

\section{Pointwise dynamic}
In this section we introduce persistent shadowable points, uniformly
$\alpha$-persistent, $\beta$-persistent points for a continuous
action $\varphi$. Also, we recall notions of shadowable points,
$\alpha$-persistent points and $\beta$-persistent points for
continuous action $\varphi:G\times X\to X$. This section consists of
3-subsection. In Subsection \ref{3001}, we study relations between
various of shadowable points. In Subsection \ref{400}, we study the
set of  persistent shadowable points and we give some properties of
it. Finally, in Subsection \ref{401}, we study the relation between
compatibility of a measure  with respect $P$-property and measure of
points in $X$ with $P$-property, where $P$-property can be
persistent shadowing property, shadowing property,
$\alpha$-persistent and $\beta$-persistent.

\subsection{Relation between various shadowable
points}\label{3001}
\begin{definition}
Let $S$ be a finitely generating set of $G$ and  $\varphi:G\times
X\to X$ be a continuous action.
\begin{enumerate}
\item (\cite{sang}) A point  $x\in X$ is called shadowable point  for $G$- action
$\varphi:G\times X\to X$, if for every $\epsilon>0$ there is
$\delta=\delta(\epsilon,x)>0$ such that for every $\delta$- pseudo
orbit $f:G\to X$ with $f(e)=x$ there is $p\in X$ such that $d(f(g),
\varphi(g, p))<\epsilon$ for all $g\in G$.
\item  A point $x\in X$ is $\alpha$-persistence  ( uniformly $\alpha$-persistent) for a continuous action $\varphi:G\times X\to X$
if for every $\epsilon>0$ there is $\delta_x>0$ such that  for every
continuous action $\psi:G\times X\to X$ with $d_S(\varphi,
\psi)<\delta$ ( and every $x'\in B_\delta(x)$)  there is $y\in X$
such that $d(\varphi(g, y), \psi(g, x))<\epsilon$ ( resp.
$d(\varphi(g, y), \psi(g, x'))<\epsilon$ ) for all $g\in G$.
Hereafter $Perssis_\alpha(\varphi)$ and $UPersis_\alpha(\varphi)$
will denote the set of all $\alpha$-persistence points  and
uniformly $\alpha$-persistent points  of $\varphi$, respectively.
 \item  A point $x\in X$ is $\beta$-persistence ( uniformly $\beta$-persistent) for a continuous action $\varphi:G\times X\to X$
if for every $\epsilon>0$ there is $\delta_x>0$ such that  for every
continuous action $\psi:G\times X\to X$ with $d_S(\varphi,
\psi)<\delta$ ( and every $x'\in B_\delta(x)$)  there is $y\in X$
such that $d(\varphi(g, x), \psi(g, y))<\epsilon$ ( resp.
$d(\varphi(g, x'), \psi(g, y))<\epsilon$) for all $g\in G$.
Hereafter $Perssis_\beta(\varphi)$ and $UPersis_\beta(\varphi)$ will
denote the set of all $\beta$-persistence points  and uniformly
$\beta$-persistent points  of $\varphi$, respectively.
\item  A point
$x\in X$ is called persistent  shadowable point for
$\varphi:G\times X\to X$, $x\in PSh(\varphi)$,  if for every
$\epsilon>0$ there is $\delta>0$ such that every $\delta$-pseudo
orbit $f:G\to X$ for $\psi:G\times X\to X$ with $d_S(\varphi,
\psi)<\delta$  and $f(e)=x$ can be $(\psi, \epsilon)$-shadowed by a
point.

\end{enumerate}

\end{definition}

It is easy to see that $PSh(\varphi)\subseteq Sh(\varphi)\subseteq
UPersis_\alpha(\varphi\subseteq Persis_\alpha(\varphi)$ and
$PSh(\varphi)\subseteq UPersis_\beta(\varphi)\subseteq
Persis_\beta(\varphi)$. The following example shows that the
converse of inclusions need not be hold.

\begin{example}\label{non-shadowable}

 \begin{enumerate}
   \item $Persis_\alpha\neq UPersis_\alpha(\varphi)$. Let $X=\mathbb{S}^1\cup\{(x, 0): -1\leq x\leq 1\}$. Since  $(-1, 0)$ is
a fixed point of every continuous action, then  $(1, 0)\in
Persis_\alpha(\varphi)$ where $\varphi:F_2\times X\to X$ is defined
by $\varphi(g, x)=x$. We claim that $(1, 0)\in
Persis_\alpha(\varphi)- UPersis_\alpha(\varphi)$. By contradiction,
we assume that $(1, 0)\in UPersis_\alpha(\varphi)$. For $\epsilon>0$
there is $\delta>0$ by $(1, 0)\in UPersis_\alpha(\varphi)$.
 Let $s_1:[-1,1]\to\mathbb{R}:x\mapsto\frac{\delta}{4}(1-|x|)$ and $s_2:[-1,1]\to\mathbb{R}:x\mapsto\frac{\delta}{6}(1-|x|)$  then $-1\le x-s_i(x)\le x+s_i(x)\le 1$ for any $x\in[-1,1]$, and $s_i(-1)=s_i(1)=0$, for $i=1, 2$. Define

$$g_i:X\to X:\langle x,y\rangle\mapsto\begin{cases}

\langle x-s_i(x),0\rangle,&\text{if }y=0\\

\left\langle x,y\right\rangle,&\text{otherwise}

\end{cases}$$

 For any $(x, y)\in \{(x, 0): -1<x<1 \}$, the first coordinate of $g_i((x, y))$  is less than $x$.
 Assume that $\psi:F_2\times X\to X$ is generated by $\varphi_a=g_1$ and $\varphi_b=g_2$. Then $d_S(\varphi, \psi)<\delta$, hence for
 $y\in (-1, 1)\times \{0\}$ with $d(y, (1, 0))<\delta$, there is $p\in X$ with $d(p, \psi(g, y))<\epsilon$ for all $g\in G$ that is a contradiction, because $g^k_i(y)\to (-1, 0)$ as $k\to \infty$.
   \item By \cite[Remark 4.4]{art1}, there is system $(X, f)$ such that $f:X\to X$ is $\alpha$-persistent while it does not shadowing property. Hence $UPersis_\alpha(f)=X$ but $Sh(f)\neq X$.
 \end{enumerate}
\end{example}

A point $x\in X$ is an equicontinuous point for continuous action
$\varphi:G\times X\to X$ if for every $\epsilon>0$ there is
$\delta_x>0$ such that
\begin{equation}
d(x, y)<\delta_x\Rightarrow d(\varphi(g, x), \varphi(g,
y))<\epsilon, \forall g\in G
\end{equation}
The set of equicontinuous points for $\varphi:G\times X\to X$ is
denoted by $Eq(\varphi)$.  It is easy to see that
\begin{equation}
Persis_\beta(\varphi)\cap Eq(\varphi)\subseteq
UPersis_\beta(\varphi)
\end{equation}
and

\begin{equation}\label{268}
Persis_\alpha(\varphi)\cap Eq(\varphi)\subseteq
Persis_\beta(\varphi)
\end{equation}

 Let $\varphi:G\times X\to X$ be continuous action as in
Example\ref{non-shadowable}. Then $\varphi$ is equcontinuous action
and $(1, 0)\in Persis_\beta(\varphi)=UPersis_\beta(\varphi)$ while
by Example \ref{non-shadowable}, $(1, 0)\notin
UPersis_\alpha(\varphi)$. This implies that
$UPersis_\beta(\varphi)\neq UPersis_\alpha(\varphi)$ and the
converse of Relation \ref{268} does not hold.\\
In the following, we show that on compact manifold $M$ without
boundary with $dim(M)\geq 2$, notions of shadowable point, uniform
$\alpha$-persistent point and $\alpha$-persistent point are
equivalent. It is not hard to see that $x$ is shadowable point if
and only if it is   finite shadowable point. We say that $x\in X$ is
finite shadowable point, if for every $\epsilon>0$ there is
$\delta>0$ such that for every $n\in \mathbb{N}$, every $\delta-n$-
pseudo orbit $f:G_n\to X$ with $f(e)=x$ can be $\epsilon$-shadowed
by point $p\in X$. Note that $G_n=\{g\in G: |g|_S\leq n\}$.

  \begin{definition}
We say that the space $X$ is generalized homogeneous, if for every
$\epsilon>0$ there exists $\delta > 0$ such that if $\{(x_1,
y_1),\ldots , (x_n, y_n)\}$ is a finite set of points in $X\times X$
satisfying:
\begin{enumerate}
    \item for every $i=1,\ldots,n, d(x_i,y_i)<\delta$,
    \item if $i\neq j$ then $x_i\neq x_j$ and $y_i\neq y_j$,
\end{enumerate}
then there is a homeomorphism $h:X\rightarrow X$ with $d_0(h,
id)<\epsilon$ and $h(x_i)=y_i$ for $i=1,\ldots,n$. \label{homogen}
\end{definition}
For example, a topological manifold $X$ without  boundary
($dim(X)\geq 2$), a Cartesian product of a countably infinite number
of manifolds with nonempty boundary and a cantor set are generalized
homogeneous \cite{pil}.
\begin{proposition}\label{u}
Let $X$ be a generalized homogeneous  compact metric space and
$\varphi:G\times X\rightarrow X$ be a continuous action. Then
$Sh(\varphi)= Persis_\alpha(\varphi)$.
\end{proposition}
\begin{proof}
  It is clear that $Sh(\varphi)\subseteq Persi_\alpha(\varphi)$. Let $x\in Persis_\alpha(\varphi)$ and $\epsilon$ be given. W claim that there is $\delta>0$ such that for every $n\in \mathbb{N}$, every $\delta-n$- pseudo orbit $f:G_n\to X$ with $f(e)=x$ can be $\epsilon$-shadowed by point $p\in X$. Choose $0<\epsilon_0<\frac{\epsilon}{2}$ corresponding to $\epsilon>0$ by $x\in Persis_\alpha(\varphi)$. Choose $0<\delta_0<\frac{\epsilon_0}{2}$ corresponding to $\epsilon_0>0$ by Definition \ref{homogen}. For $\delta_0>0$ there is $0<\delta<\frac{\delta_0}{2}$ such that
   \begin{equation*}
     d(a, b)<\delta\Rightarrow d(\varphi(g, a), \varphi(g, b))<\delta_0, \forall |g|_S\leq 2.
   \end{equation*}

   Let $f:G_n\to X$ be a $\delta-n$pseudo orbit with $f(e)=x$. Similar proof of Lemma 2.1.2 in \cite{pil}, we can construct $\delta_0$-pseudo orbit $F:G_n\to X$ with the following property:
   \begin{itemize}
     \item $F(e)=x$,
     \item $\varphi(s, F(g))\neq F(sg), \forall s\in S, \forall |g|_S<n$,
     \item $d(F(g),f(g))<\delta_0, \forall g\in G_n$
   \end{itemize}

   By Definition \ref{homogen}, for $\{ (\varphi(s, F(g)), F(sg)); g\in G_n\}$, there is a homeomorphism  $h: X\to X$ with $d_0(h, id)<\epsilon_0$ such that $h(\varphi(s, F(g)))= F(sg)$. Let $\psi:G\times X\to X$ be generated by $h\circ \varphi_s$ for $s\in S$. Then $\psi:G\times X\to X$ is $\epsilon_0$-close to $\psi:G\times X\to X$. This implies that there is $p\in X$ with $d(\psi(g, x), \varphi(g, p))<\epsilon$ for all $g\in G$. But $\psi(g, x)=F(g)$, hence $d(f(g), \varphi(g, p))\leq d(f(g), F(g))+ d(F(g), \varphi(g, p))<\epsilon$ for all $g\in G$.
\end{proof}

By Theorem 3.5 in \cite{sang}, a continuous action $\varphi:G\times
X\to X$ on compact metric space $(X, d)$ has shadowing property if
and only if it is pointwise shadowable point. Hence by Proposition
\ref{u}, we have

\begin{corollary}
 If $\varphi:G\times X\to X$ is a continuous action of finitely generated group $G$ on generalized homogeneous  compact metric space $(X, d)$, then the following equivalent:
 \begin{enumerate}
   \item $\varphi$ has shadowing property,
   \item $\varphi$ is pointwise shadowable point,
   \item $\varphi$ is pointwise $\alpha$-persistent,
   \item $\varphi$ is $\alpha$-persistent.
 \end{enumerate}
 \end{corollary}
Let $X$ be a compact metric space. It is easy to see that the
following properties hold.
\begin{enumerate}
    \item Continuous action $\varphi:G\times X\to X$ is
    $\alpha$-persistent if and only if $UPersis_\alpha(\varphi)=X$
    \item Continuous action $\varphi:G\times X\to X$ is
    $\beta$-persistent if and only if $UPersis_\beta(\varphi)=X$
    \item Equicontinuous action $\varphi:G\times X\to X$ is
    $\beta$-persistent if and only if $Persis_\beta(\varphi)=X$.

\end{enumerate}

\subsection{Some properties of persistent shadowable
points}\label{400}

In the following, we give some properties of persistent shadowable
points.

\begin{theorem}\label{wok}
  Let $S$ be a finitely generating set of $G$ and $\varphi:G\times X\to X$ be a continuous action on compact metric space $(X, d)$.
  \begin{enumerate}
    \item The point $x=p$ is persistent shadowable if and only if for every $\epsilon>0$ there is $\delta>0$ such that every $\delta$-pseudo orbit through $B[p, \delta]$ of a continuous action $\psi:G\times X\to X$ with $d_S(\varphi, \psi)<\delta$ can be shadowed by a $\psi$-orbit.
    \item Continuous action $\varphi:G\times X\to X$ has persistent shadowing property on compact set $K$ if and only if $K\subseteq PSh(\varphi)$.
    \item Continuous action $\varphi:G\times X\to X$  has the persistent shadowing property if and only if it is pointwise persistent shadowable.
    \item $PSh(\varphi)= UPersis_\beta(\varphi)\cap Sh(\varphi)$.
   \item Continuous action $\varphi:G\times X\to X$ has persistent shadowing property if and only if it is $\beta$-persistent and it has shadowing property.
  \end{enumerate}
\end{theorem}
\begin{proof}
  \begin{enumerate}
  \item   Suppose by contradiction that $x=p$ is persistent shadowable point but there are $\epsilon>0$, a sequence of continuous actions $\psi_k:G\times X\to X$ with $d_S(\varphi, \psi_k)\leq \frac{1}{k}$, a sequence of $\frac{1}{k}$-pseudo orbits $f^k:G\to X$ of $\psi_k:G\times X\to X$ with $d(f^k(e), x)\leq \frac{1}{k}$ such that $f^k:G\to X$ can not be $2\epsilon$-shadowed by any $\psi_k$-orbit, for every $k\in \mathbb{N}$. For this $\epsilon$ we let $\delta$ be given by the persistently shadowableness of $x$. We can assume that $\delta<\epsilon$. \\
  On the one hand,
  \begin{align*}
     d(\psi_k(s, p), \psi_k(s, f^k(e)) & \leq d(\psi_k(s,p), \varphi(s, p))+ d(\varphi(s, p), \varphi(s, f^k(e)))+ d(\varphi(s, f^k(e)), \psi_k(s, f^k(e)))\\
     & \leq 2d_s(\varphi, \psi_k)+ d(\varphi(s, p), \varphi(s, f^k(e)))
  \end{align*}
  We can choose $k$ large satisfying
  \begin{equation}\label{301}
    \max \{d(\psi_k(s, p), \psi_k(s, f^k(e)), \frac{1}{k}\}<\frac{\delta}{2}, \forall s\in S
  \end{equation}

Let us define $F^k:G\to X$ by
 \[ F^k(g)= \left\lbrace
  \begin{array}{c l}
     f^k(g), & \text{\rm{ $ g\neq e$}},\\
e, & \text{\rm{$g=e$}}.
  \end{array}
\right. \]

Then

 \[ d(\psi_k(s, F^k(g)), F^k(sg))= \left\lbrace
  \begin{array}{c l}
     d(\psi_k(s, f^k(g)), f^k(sg), & \text{\rm{ $ \text{ for } g\notin\{e, s^{-1}: s\in S\}$}},\\
d(\psi_k(s, f^k(g)), p), & \text{\rm{$ \text{ for } g=s^{-1}$}},\\
d(\psi_k(s, p), f^k(s)), & \text{\rm{$ \text{ for } g=e$}}.
  \end{array}
\right. \]

Since $f^k:G\to X$ is $\frac{1}{k}$-pseudo orbit of $\psi_k:G\times
X\to X$, hence
\begin{equation}\label{zx}
  d(\psi_k(s, F^k(g)), F^k(sg))<\frac{1}{k}<\delta, \text{ for } g\notin \{e, s^{-1}: s\in S\}
\end{equation}
Also inequality
\begin{equation*}
  d(\psi_k(s, f^k(s^{-1})), p)\leq d(\psi_k(s, f^{k}(s^{-1})), f^{k}(e))+d(f^k(e), p),
\end{equation*}
implies that

\begin{equation}\label{zc}
  d(\psi_k(s, F^k(g)), F^k(sg))<\delta, \text{ for } g=s^{-1}.
\end{equation}
By Relation \ref{301} and Relation \ref{zc}
\begin{equation*}
  d(\psi_k(s, p), f^k(s))\leq d(\psi_k(s, p)+\psi_k(s, f^k(e)))+ d(\psi_k(s, f^k(e)), f^k(s))
\end{equation*}
we have
\begin{equation}\label{zn}
  d(\psi_k(s, F^k(g)), F^k(sg))<\delta, \text{ for } g=e
\end{equation}
Hence by Relations \ref{zx},\ref{zc}, \ref{zn}, we get $F^k:G\to X$
is $\delta$-pseudo orbit of $\psi_k:G\times X\to X$. But
$d_S(\varphi, \psi_k)<\delta$, hence , by persistent shadowing
property of $\varphi$, for $\delta$-pseudo orbit $F^k:G\to X$ with
$F(e)=p$ of continuous action $\psi_k$ with $d_S(\varphi,
\psi_k)<\delta$, there is $z\in X$ such that $d(F^k(g), \psi_k(g,
z))<\epsilon$ for all $g\in G$. Since for $g\neq e$ one has
$d(f^k(g), \psi_k(g, z))=d(F^k(g), \psi_k(g, z))<\epsilon$ and for
$g=e$
\begin{equation*}
  d(z, f^k(e))\leq d(z, p)+ d(p, f^k(e))<\epsilon+\frac{1}{k}<2\epsilon,
\end{equation*}
Hence  $f^k:G\to X$ can be $2\epsilon$- shadowed by $\psi_k$-orbit
of $z\in X$. That is a contradiction.
\item It is sufficient to show that if $K\subseteq PSh(\varphi)$, then $\varphi$ has persistent shadowing property on $K$. Let $\epsilon>0$ be given. For every $x\in X$ there is $\delta_x>0$ corresponding to $\epsilon>0$ by item (2). Since $K$ is a compact space, the open cover $\{B[x, \delta_x]:x\in K\}$ has a finite open subcover $\{B[x_i, r_{x_i}]: i=1, 2, \ldots, n\}$.\\
  Take $\delta=\min \{ \delta_{x_i}: i\in \{1, 2, \ldots, n\}\}$ and let $F:G\to X$ be a $\delta$-pseudo orbit of continuous action $\psi:G\times X\to X$ with $d_S(\varphi, \psi)<\delta$ and $F(e)\in K$. Clearly $F(e)\in B[x_i, \delta_{x_i}]$ for some $1\leq i\leq n$. This implies that $F:G\to X$ is $\delta_{x_i}$-pseudo orbit through $B[x_i, \delta_{x_i}]$. Then $F:G\to X$ can be eventually $\epsilon$-shadowed by some $\psi$-orbit.
  \item Take  $K = X$ in item (3), we have that $\varphi$ has the
persistent shadowing property if and only if $PSh(\varphi)=X$.
   \item   Firstly, we show that $PSh(\varphi)\subseteq UPersis_\beta(\varphi)\cap Sh(\varphi)$. Take $x\in PSh(\varphi)$, $\epsilon>0$. Choose  $\delta_0>0$ corresponding $\frac\epsilon2>0$ by $x\in PSh(\varphi)$. Choose $\delta<\frac{\delta_0}{2}$ such that
       \begin{equation*}
         d(a, b)<\delta\Rightarrow d(\varphi(s, a), \varphi(s, b))<\frac{\delta_0}{2}.
       \end{equation*}
       Fix a continuous action $\psi:G\times X\to X$ with $d_S(\varphi, \psi)<\delta$. For $y\in B_\delta(x)$, define
       $F:G\to X$ by $F(g)=\varphi(g, y)$ if $g\neq e$ and $F(e)=x$, then $F:G\to X$ is a $\delta$-pseudo orbit of continuous action $\psi:G\times X\to X$ with $F(e)=x$. By $x\in PSh(\varphi)$, there is $p\in X$ such that $d(F(g), \psi(g, p))<\frac\epsilon2$ for every $g\in G$. This implies that $d(\varphi(g, y), \psi(g, p))<\epsilon$ for all $g\in G$. It follows that $x\in UPersis_\beta(\varphi)$. Since $PSh(\varphi)\subseteq Sh(\varphi)$, we get $x\in UPersis_\beta(\varphi)\cap Sh(\varphi)$. Therefore, $PSh(\varphi)\subseteq UPersis_\beta(\varphi)\cap Sh(\varphi)$.

       Now we show that $UPersis_\beta(\varphi)\cap Sh(\varphi)\subseteq PSh(\varphi)$. Suppose that   $x\in UPersis_\beta(\varphi)\cap Sh(\varphi)$ and  $\epsilon>0$ be given. We show that there is $\delta>0$ such that for every $\delta$-pseudo orbit $f:G\to X$ with $f(e)=x$ of $\psi$ such that $d_S(\varpi, \psi)<\delta$, there is $p\in X$ with $d(f(g), \psi(g, p))<\epsilon$ for all $g\in G$.
        Choose $\epsilon_0<\frac\epsilon4$ corresponding $\frac\epsilon2$ by $x\in UPersis_\beta(\varphi)$. There is $\eta>0$ corresponding $\frac{\epsilon_0}{2}$ by $x\in Sh(\varphi)$. Take $\delta<\frac\eta2$. If $f:G\to X$ is $\delta$-pseudo orbit of $\psi$ with $f(e)=x$ for continuous action $\psi:G\times X\to X$ with $d_S(\varphi, \psi)<\delta$, then $f:G\to X$ is $\eta$-pseudo orbit of $\varphi$ with $f(e)=x$. By $x\in Sh(\varphi)$ and $f(e)=x$, there is $y\in B_{\epsilon_0}(x)$ such that $d(f(g), \varphi(g, y))<\frac\epsilon2$ for all $g\in G$. Also by  $x\in UPersis_\beta(\varphi)$ and $y\in B_{\epsilon_0}(x)$ there is $p\in X$ such that $d(\varphi(g, y), \psi(g, p))<\frac\epsilon2$ for all $g\in G$. This implies that $d(f(g), \psi(g, p))<\epsilon$ for all $g\in G$.
.
   \item It is clear that persistent shadowing property implies $\beta$-persistent and shadowing property. For the converse let $\varphi$ is $\beta$-persistent
    and it has shadowing property, then by item (4), $PSh(\varphi)= Persis_\beta(\varphi)\cap Sh(\varphi)= X$, hence $\varphi$ is pointwise persistent shadowple and by item (3), $\varphi$ has persistent shadowing property.
\end{enumerate}
\end{proof}
\subsection{Various shadowable points and related
measures}\label{401}
 Item 1 in Proposition \ref{wok} implies that if
continuous action $\varphi:G\times X\to X$ does have persistent
shadowing property on compact set $K\subseteq X$, then for every
$\epsilon>0$ there
 exist a neighborhood $U$ of $K$ and  $\delta>0$ such that $U\subseteq PSh_{\varphi}(\delta, \epsilon)$. Also, by Item 2 in Proposition \ref{wok}, $K\subseteq PSh(\varphi)$
 implies that  continuous action
$\varphi:G\times X\to X$ does have persistent  shadowing property on
compact set $K\subseteq X$. Hence we have the following proposition.

\begin{proposition}\label{rtg}
Let $\varphi:G\times X\to X$  be a continuous action on compact
metric space $(X, d)$. Let $K\subset PSh(\varphi)$ be a compact
subset. Then for every $\epsilon>0$ there exist a neighborhood $U$
of $K$  and $\delta>0$  such that $U\subseteq PSh_{\varphi}(\delta,
\epsilon)$.

\end{proposition}

 One can check that Proposition \ref{rtg} is true in the case of
 shadowing property, $\alpha$-persistent and $\beta$-persistent.

 \begin{proposition}
 If $\mathcal{X}\in \{ Sh(\varphi),
 UPersis_\alpha(\varphi), UPersis_\beta(\varphi)\}$ and $K\subseteq
 X$ be a compact set. Then for every $\epsilon>0$ there exist a
 neighborhood $U$ of $K$ and $\delta>0$ such that $U\subseteq
 \mathcal{X}_{\varphi}(\delta, \epsilon)$.
 \end{proposition}
Assume that $supp(\mu)\subset PSh(X, \varphi)$ and $X$ be a compact
metric space. Since $supp(\mu)$ is a compact set, hence by
Proposition \ref{rtg}, for every $\epsilon>0$ there is $\delta>0$
such that
\begin{equation}
A\cap supp(\mu)\neq\emptyset \Rightarrow A\cap PS_{\delta,
\epsilon}(X, \varphi)\neq\empty.
\end{equation}
Hence we have the following corollary
\begin{equation}\label{pkiii}
supp(\mu)\subseteq PSh(\varphi)\Rightarrow
\mu\in\mathcal{M}_{PSh}(X, \varphi)
\end{equation}
By Proposition \ref{pki} and Remark \ref{pkii} and with similar
proof of Remark \ref{pkiii} we have the following proposition.
\begin{proposition}\label{Lb}
Let $\varphi:G\times X\to X$ be a continuous action of finitely
generated group on compact metric space $(X, d)$. Then
\begin{enumerate}
    \item $\mu\in M_{PSh}(X, \varphi)\Leftrightarrow
    supp(\mu)\subseteq PSh(\varphi)$,
    \item $\mu\in M_{Sh}(X, \varphi)\Leftrightarrow
    supp(\mu)\subseteq Sh(\varphi)$,
    \item $\mu\in M_{\alpha}(X, \varphi)\Leftrightarrow
    supp(\mu)\subseteq UPersis_\alpha(\varphi)$,
    \item $\mu\in M_{\beta}(X, \varphi)\Leftrightarrow
    supp(\mu)\subseteq UPersis_\beta(\varphi)$
\end{enumerate}
\end{proposition}
By Proposition \ref{kj}, the set of persistent shadowable points is
measureable. With similar proof, one can check that $Sh(\varphi)$,
$UPersis_\alpha(\varphi)$ and $UPersis_\beta(\varphi)$ are
measureable sets. Assume that $supp(\mu)\subseteq
\overline{PSh(\varphi)}$, then by Lemma 2.8 in \cite{boom}, if $X$
is a compact metric space, then there is a sequence
$\mu_n\in\mathcal{M}(X)$ with $supp(\mu_n)\subseteq PSh(\varphi)$
and converging to $\mu$ with respect to the $weak^*$ topology. By
Proposition \ref{Lb}, $\mu_n\in\mathcal{M}_{PSh}(X, \varphi)$. This
implies that $\mu\in \overline{\mathcal{M}_{PSh}(X, \varphi)}$.
Hence, we have  following relation
\begin{equation}\label{27}
 \mu(\overline{PSh(\varphi)})=1\Rightarrow \mu\in\overline{\mathcal{M}_{PSh}(X, \varphi)}.
\end{equation}
Conversely, let $\mu\in\overline{\mathcal{M}_{PSh}(X, \varphi)}$.
Choose $\mu_n\in \mathcal{M}_{PSh}(X, \varphi)$ such that $\mu_n\to
\mu$.  By inequality
$$\limsup_{n\to\infty}\mu_n(\overline{PSh(\varphi)})\leq
\mu(\overline{PSh(\varphi)})$$ and
$\mu_n(\overline{PSh(\varphi)})=1$, we have
$\mu(\overline{PSh(\varphi)})=1$. Hence we have the following
relation.
\begin{equation}\label{277}
\mu\in\overline{\mathcal{M}_{PSh}(X, \varphi)}\Rightarrow
\mu(\overline{PSh(\varphi)})=1.
\end{equation}
By Relation \ref{27} and Relation \ref{277}, we have the following
proposition.
\begin{proposition}\label{278}
Let $\varphi:G\times X\to X$ be a continuous action on compact
metric space $(X, d)$. Then $\mu(\overline{PSh(\varphi)})=1$ if and
only if $\mu\in\overline{\mathcal{M}_{PSh}(X, \varphi)}$.
\end{proposition}
With similar proof, we have the following proposition.
\begin{proposition}\label{Lbb}
Let $\varphi:G\times X\to X$ be a continuous action on compact
metric space $(X, d)$. Then
\begin{enumerate}
    \item $\mu(\overline{PSh(\varphi)})=1\Leftrightarrow \mu\in\overline{\mathcal{M}_{PSh}(X,
    \varphi)}.$
    \item $\mu(\overline{Sh(\varphi)})=1\Leftrightarrow \mu\in\overline{\mathcal{M}_{Sh}(X,
    \varphi)}.$
    \item $\mu(\overline{UPersis_\alpha(\varphi)})=1\Leftrightarrow \mu\in\overline{\mathcal{M}_\alpha(X,
    \varphi)}.$
    \item $\mu(\overline{UPersis_\beta(\varphi)})=1\Leftrightarrow \mu\in\overline{\mathcal{M}_\beta(X,
    \varphi)}.$
\end{enumerate}
\end{proposition}
 We claim that if
$PSh(\varphi)$ is a closed set in $X$, then
$\overline{\mathcal{M}_{PSh}(X,
    \varphi)}= \mathcal{M}_{PSh}(X,
    \varphi)$. Take $\mu_n\in\mathcal{M}_{PSh}(X, \varphi)$ with
    $\mu_n\to \mu$. Then $\limsup_{n\to
    \infty}\mu_n(PSh(\varphi))\leq
    \mu(PSh(\varphi))$ implies that
    $\mu(PSh(\varphi))=1$. Hence $supp(\mu)\subseteq
    PSh(\varphi)$. By Proposition \ref{Lb},
    $\mu\in\mathcal{M}_{PSh}(\varphi)$ i.e.
\begin{equation}\label{2312}
    \overline{PSh(\varphi)}=PSh(\varphi)\Rightarrow\overline{\mathcal{M}_{PSh}(X, \varphi)}= \mathcal{M}_{PSh}(X, \varphi).
\end{equation}

Conversely, let $\overline{\mathcal{M}_{PSh}(X,
    \varphi)}= \mathcal{M}_{PSh}(X,
    \varphi)$, we claim that $\overline{PSh(\varphi)}=PSh(\varphi)$.
    If it is not true, then there exist $\{x_n\}\subseteq
    PSh(\varphi)$ with $x_n\to x$ such that $x\notin PSh(\varphi)$.
    By $x_n\to x$, we have $m_{x_n}\to m_x$.
 Where  $m_t$ the Dirac measure supported on $t\in X$, indeed,
$m_t(A)=0$ or $1$ depending on whether $t\notin A$ or $t\in A$. It
is easy to see that

$$PSh(\varphi)=\{t\in X:m_t\in\mathcal{M}_{PSh}(X, \varphi)\}$$
 By $\{x_n\}\subseteq
    PSh(\varphi)$, we have $m_{x_n}\in\mathcal{M}_{PSh}(X,
    \varphi)$. Also, by $m_{x_n}\to m_x$ and $\overline{\mathcal{M}_{PSh}(X,
    \varphi)}= \mathcal{M}_{PSh}(X,
    \varphi)$, we have $x\in PSh(\varphi)$ which is a contradiction.
    This implies the following relation.

    \begin{equation}\label{12321}
\overline{\mathcal{M}_{PSh}(X,
    \varphi)}= \mathcal{M}_{PSh}(X,
    \varphi)\Rightarrow \overline{PSh(\varphi)}=PSh(\varphi).
    \end{equation}
    By Relation \ref{2312} and Relation \ref{12321}, we have the
    following proposition.
    \begin{proposition}\label{12354}
    Let $\varphi:G\times X\to X$ be a continuous action on compact
    metric space $(X, d)$. Then
    $$\overline{\mathcal{M}_{PSh}(X,
    \varphi)}= \mathcal{M}_{PSh}(X,
    \varphi)\Leftrightarrow \overline{PSh(\varphi)}=PSh(\varphi).$$
    \end{proposition}

 On can check that result of Proposition \ref{12354} can be obtain
 for other type of shadowing,indeed we have the following
 proposition.

   \begin{proposition}\label{12355}
    Let $\varphi:G\times X\to X$ be a continuous action on compact
    metric space $(X, d)$. Then
    \begin{enumerate}
        \item $\overline{\mathcal{M}_{PSh}(X,
    \varphi)}= \mathcal{M}_{PSh}(X,
    \varphi)\Leftrightarrow \overline{PSh(\varphi)}=PSh(\varphi).$
        \item $\overline{\mathcal{M}_{Sh}(X,
    \varphi)}= \mathcal{M}_{Sh}(X,
    \varphi)\Leftrightarrow \overline{Sh(\varphi)}=Sh(\varphi).$
        \item $\overline{\mathcal{M}_{\alpha}(X,
    \varphi)}= \mathcal{M}_{\alpha}(X,
    \varphi)\Leftrightarrow \overline{Persis_\alpha(\varphi)}=Persis_\alpha(\varphi).$
        \item $\overline{\mathcal{M}_{\beta}(X,
    \varphi)}= \mathcal{M}_{\beta}(X,
    \varphi)\Leftrightarrow \overline{Persis_\beta(\varphi)}=Persis_\beta(\varphi).$
    \end{enumerate}

    \end{proposition}

If  $\varphi:G\times X\to X$ is equicontinuous
    action, then $UPersis_\beta(\varphi)=Persis_\beta(\varphi)$
is a closed subset of $X$. This implies the following proposition.

\begin{proposition}\label{cin}
Let $\varphi:G\times X\to X$ be a equicontinuous action of a
finitely generated  group $G$ on compact metric space. Then
\begin{enumerate}
    \item $\overline{\mathcal{M}_{\beta}(X,
    \varphi)}= \mathcal{M}_{\beta}(X,
    \varphi)$
    \item $\mu(Persis_\beta(\varphi))= 1$ if and only if $\mu\in\mathcal{M}_\beta(X,
    \varphi)$.
\end{enumerate}
\end{proposition}
 Proof of the following proposition is clear.
\begin{proposition}\label{diracm}
Let $\varphi:G\times X\to X$ be a continuous action.
\begin{enumerate}
    \item $PSh(\varphi)=\{x\in X: m_x\in\mathcal{M}_{PSh}(X, \varphi)\}$.
    \item $Sh(\varphi)=\{x\in X: m_x\in\mathcal{M}_{Sh}(X, \varphi)\}$
    \item $Persis_\beta(\varphi)=\{x\in X: m_x\in\mathcal{M}_{\beta}(X, \varphi)\}$
    \item $Persis_\alpha(\varphi)=\{x\in X: m_x\in\mathcal{M}_{\alpha}(X, \varphi)\}$
\end{enumerate}

\end{proposition}
Assume that $\overline{PSh(\varphi)}=X$. By Theorem 6.3 in
\cite{par}, if $X$ is a separable metric space, then Then the set of
all measures whose supports are finite subsets of $PSh(\varphi)$ is
dense in $\mathcal{M}(X)$. Also by Lemma 2.7 in \cite{boom}, every
measure with finite support and supported on $PSh(\varphi)$ is a
finite convex combination of Dirac measures supported on points of
$PSh(\varphi)$. By Proposition \ref{diracm}, such Dirac measures are
compatible with persistent shadowing property. But finite convex
combination of measures in $\mathcal{M}_{PSh}(X, \varphi)$ is
compatible with persistent shadowing property. This implies that if
$\overline{PSh(\varphi)}=X$, then the set of finite convex
combination of measures in $\mathcal{M}_{PSh}(X, \varphi)$ is dense
in $\mathcal{M}(X)$. This means that if $\overline{PSh(\varphi)}=X$,
then $\overline{\mathcal{M}_{PSh}(X,\varphi)}=\mathcal{M}(X)$.
Conversely, assume that
$\overline{\mathcal{M}_{PSh}(X,\varphi)}=\mathcal{M}(X)$. We claim
that $\overline{PSh(\varphi)}=X$. If it is not true, then there is
$x\in X$ with $x\notin \overline{PSh(\varphi)}$. Choose open set $U$
of $x$ with $x\in U\subseteq X-\overline{PSh(\varphi)}$. Since
$m_x\in \overline{\mathcal{M}_{PSh}(X, \varphi)}$, there is a
sequence $\mu_n\in\mathcal{M}_{PSh}(X, \varphi)$ such that $\mu_n\to
m_x$. By Proposition \ref{Lb}, we have $Supp(\mu_n)\subseteq
PSh(\varphi)$. By $ U\subseteq X-\overline{PSh(\varphi)}$, we have
$\mu_n(U)=0$ for all $n\in \mathbb{N}$. There for $0=\liminf_{n\to
\infty}\mu_n(U)\geq m_x(U)=1$ which is a contradiction. With similar
technics we can prove the following propositions.
\begin{proposition}\label{3214}
Let $\varphi:G\times X\to X$ be a continuous action of a finitely
generated group $G$ on compact metric space $(X, d)$. Then the
following conditions hold.
\begin{enumerate}
    \item $\overline{\mathcal{M}_{PSh}(X, \varphi)}=\mathcal{M}(X)
    \Leftrightarrow \overline{PSh(\varphi)}=X$
    \item $\overline{\mathcal{M}_{Sh}(X, \varphi)}=\mathcal{M}(X)
    \Leftrightarrow \overline{Sh(\varphi)}=X$
    \item $\overline{\mathcal{M}_\beta(X, \varphi)}=\mathcal{M}(X)
    \Leftrightarrow \overline{Persis_\beta(\varphi)}=X$
    \item $\overline{\mathcal{M}_{\alpha}(X, \varphi)}=\mathcal{M}(X)
    \Leftrightarrow \overline{Persis_\alpha(\varphi)}=X$
\end{enumerate}
\end{proposition}

\section*{acknowledgments} The author wishes to thank Professor Morales   for his idea about Theorem \ref{wok} given in \cite{morales2}.

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


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text:
UN chief warns Syria crisis is getting worse
Published on April 6, 2012 at 9:40 am
UNITED NATIONS (AP) — Secretary-General Ban Ki-moon said Thursday the crisis in Syria is getting worse and claiming more lives every day even though President Bashar Assad's government insists it is withdrawing troops ahead of a U.N. deadline to end the violence.
The U.N. chief appealed to Assad "to show vision and leadership" and keep his pledge to pull troops and heavy weapons out of cities and towns by April 10, and he urged the opposition to be ready to stop all violence if the Syrian government meets the deadline.
"Cities, towns and villages have been turned into war zones. The sources of violence are proliferating," Ban told the U.N. General Assembly. "The human rights of the Syrian people continue to be violated. ... Humanitarian needs are growing dramatically."
His comments came as activists reported that Syrian troops attacked the Damascus suburb of Douma, an assault they said shows that Assad is intensifying violence in the days before the April 10 deadline. His crackdown on the yearlong uprising has left at least 9,000 people dead, according to the U.N.
Earlier Thursday, a U.N. team arrived in Damascus to start technical preparations for the possible deployment of U.N. monitors for any cease-fire between Syrian troops and rebel forces.
Kofi Annan, the U.N.-Arab League envoy trying to end the conflict, said Syria has informed him of partial withdrawals from three locations — Idlib, Zabadani and Daraa — "but it is clear that more far-reaching action is urgently required."
"We must silence the tanks, helicopters, mortars and guns, and stop all other forms of violence too: sexual abuse, torture, executions, abductions, destruction of homes, forced displacement, and other such abuses, including on children," he said.

redpajama_set_name: RedPajamaCommonCrawl


text:
Customs foil currency smuggling bid
MQM-P secures landslide win against PTI in Karachi by-poll
Indonesia, Pakistan ties poised for a quantum leap, says envoy
Deny Basuki says both countries emerging economies of creative and talented people
DNA January 27, 2019
Deny Basuki says both countries emerging economies of creative and talented people. PHOTO: FILE
ISLAMABAD: Counsellor and head of cultural section Embassy of Republic of Indonesia Deny Tri Basuki has said Indonesia and Pakistan share strong socio-cultural and religious bond rooted in history.
Pakistan and Indonesia stand proudly together as two of the largest Muslim populated countries and emerging economies of creative and talented people.
He expressed these views on the occasion of a business gathering organised by tourism ministry of Indonesia in collaboration with the Indonesian embassy. A large number of stakeholders hailing from the travel and aviation industry of Pakistan attended the event.
Trade between Pakistan, Indonesia likely to surge to $9b
He said it was very pleasing to note that both the countries aspire to maintain and further develop their exemplary ties and transform the potential of these ties into increased cooperation in various spheres
Both the countries highly value their ties and encourage all-round cooperation to further strengthen the fraternal bond between the two nations.
He said in recent years, Indonesia-Pakistan bilateral relations have seen significant development in mutual beneficial cooperation. The two countries have endeavoured to expand their cooperation in the areas of trade and economy, defense cooperation, academic linkages and cultural interaction.
While talking about future prospects of bilateral cooperation Basuki said though Pakistan and Indonesia enjoy a deep friendly relationship, yet the scope is extraordinary and exceptional.
Tourism he said is generally about leisure and pleasure. But it also promotes cultural exchange and provides opportunity to people of various cultures to meet and understand each other.
He said two years ago the embassy, in its efforts to create awareness about tourism opportunities in Indonesia,produced a travelogue series in collaboration with a Pakistani Entertainment TV Channel, which went very well.
The travelogue series was a success as it helped in acquainting and attracting the people in Pakistan to the rich and diverse culture of Indonesia and also helped build tourism profile of Indonesia in Pakistan.
He added, in 2017, the Embassy under its FAMTRIP programme took 10 top tour operators of Islamabad, Lahore and Karachi to Indonesia adding recently five journalists from Pakistan's leading media were taken to Jakarta, Bandung and Bali to get to know more about Indonesia's tourism potential.
These days, tourism is one of the fastest growing industries in the world.
In addition to the opportunities for recreation and cultural exchange, tourism industry has emerged as a big source to generate employment opportunities, grow service sector and help earn foreign exchange, he concluded.
Published in The Express Tribune, January 27th, 2019.
Six Pakistani entrepreneurs graduate from UNCTAD, Alibaba's fellowship
IIUI, Indonesian varsity to collaborate
PM Imran condoles with President Widodo


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Health and Functional Status
Gerry E. Hendershot
David C. Stapleton, Mathematica Policy Research
Benjamin H. Harris, Brookings Institution
10.17848/9781441612687.ch7
UNEMPLOYMENT, DISABILITY, and INCOME SUPPORT PROGRAMS; Workers compensation and disability; Disability
Hendershot, Gerry E., Benjamin H. Harris, David C. Stapleton. 2009. "Health and Functional Status." In Counting Working-Age People with Disabilities: What Current Data Tell Us and Options for Improvement, Andrew J. Houtenville, David C. Stapleton, Robert R. Weathers II, Richard V. Burkhauser, eds. Kalamazoo, MI: W.E. Upjohn Institute for Employment Research, pp. 227-263. https://doi.org/10.17848/9781441612687.ch7