## A concept of dynamical coherence

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I'm trying to make an overview of the study of partial hyperbolicity and there is an interesting concept of dynamical coherence which appears there. Some call it mild (see the Thesis of Pablo Carrasco, Compact Dynamical Foliations 2010), some call it strong and unnatural (see the work of Amy Wilkinson and Keith Burns Dynamical coherence, accessibility and center bunching). The definition which is the most common is that local cental-unstable $E^{cu}$ and center-stable $E^{cs}$ bundles integrate to foliations $W^{cu}$ and $W^{cs}$. Let us suppose, that in a normally hyperbolic case, i.e. when we already have the $E^c$ that integrates to a foliation F, at which some diffeomorphism is hyperbolic.

My question is how the normally hyperbolic (i.e. partially hyperbolic on foliation) system could be dynamically incoherent and is this concept somewhat related to the concept of local product structure?

My question is, what is a simplest example of a normally hyperbolic foliation when $E^cu$ and $E^cs$ do not integrate to foliations? And how "often" does it happen in the world of normally hyperbolic foliations?

PS. Updated after a useful remark of Rafael Potrie, the definition of a dynamical coherence is now more precise.

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## 1 Answer

In general, it is not known if a partially hyperbolic diffeomorphism should be dynamically coherent.

There are two obstructions for integrability of the center bundle. One is that the distributions are not integrable (Frobenius conditions fails) and the other one is that the distributions may lack of diferentiability (and so uniqueness of integrability may fail).

For the first obstruction, Wilkinson noted that in diffeomorphisms with high dimensional center (i.e. Anosov automorphisms on nilmanifolds) the bracket condition fails.

In the absolute partially hyperbolic setting, for diffeomorphisms of $T^3$ dynamical coherence has been obtained by Brin-Burago and Ivanov (http://www.pdmi.ras.ru/~svivanov/papers/coherence.pdf). This has been extended to nilmanifolds by Hammerlindl and Parwani (http://arxiv.org/abs/1103.3724, http://arxiv.org/abs/1001.1029).

For pointwise partially hyperbolic systems (a weaker condition), recent examples have been constructed by Rodriguez-Hertz, Rodriguez-Hertz and Ures showing that dynamical coherence may fail even in $T^3$. On the other hand, I have recently proved that if the partially hyperbolic diffeomorphism of $T^3$ is transitive (or volume preserving) then it must be dynamically coherent (see http://www.mat.puc-rio.br/edai/textos/potrie.pdf).

Local product structure I believe has something to do with plaque-expansiveness, which allows one to show robustness of dynamical coherence thanks to the work of Hirsch-Pugh and Shub, but in general it is not enough to show the existence of an invariant foliation tangent to the center.

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Let's say there is a normally hyperbolic and plaque expansive foliation, isn't it always dynamically coherent? I'm not interested in partially hyperbolic case, since we have the Foliation stability theorem and the diffeomorphism remains normally hyperbolic. I just have the impression that all of normally hyperbolic and plaque expansive diffeomorfisms are dynamically coherent, maybe I do not feel the definition very well.. – nikutaibi Mar 19 2012 at 21:54
I believe we have slightly different definitions. If a \emph{foliation} is normally hyperbolic and plaque expansive for a diffeomorphism $f$, then $f$ is (as a partially hyperbolic diffeomorphism) robustly dynamically coherent. The problem with your second sentence is that in order to define plaque expansiveness, at least for the definition I know (of Hirsch-Pugh-Shub), one needs that $f$ be dynamically coherent. If there is no plaque expansiveness, it is open whether perturbations of a diffeomorphism having a normally hyperbolic invariant foliation is robustly dynamically coherent. – rpotrie Mar 22 2012 at 20:22
@ rpotrie Plaque Expansiveness can be (at least formally) defined whenever the center foliation exists. And this is the definition in many references. Why would we need the dynamical coherence assumption to define PE? Also does the following work: if $E^c_f$ integrates to $W^c_f$, then $W^c_f$ is normally hyperbolic. So for $g$ close to $f$, $E^c_g$ also integrates to some $W^c_g$ (close to $W^c_f$). Thank you! – Pengfei Mar 25 2012 at 14:10
@Pengfei: I know that the definition of dynamical coherence varies within the literature, but in the question, the definition given is the existence of the center foliation, so that is why I said that the definition of plaque-expansiveness required dynamical coherence. As for your second question, I believe that you need plaque-expansiveness or some other condition to get the persistence of integrability of the center direction (although there are no counterexamples for your statement). – rpotrie Mar 25 2012 at 21:10

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Ok, I believe I understand your question now. As far as I know, it is not known in all generality if integrability of the center implies integrability of the center-stable and center-unstable. However, you may find the work of Brin-Burago-Ivanov pdmi.ras.ru/~svivanov/papers/bbi-parthyp.pdf (see Page 4 and Proposition 3.4) and the following work by Burago and Ivanov pdmi.ras.ru/~svivanov/papers/foliations.pdf (Proposition 3.1) helpful. They prove that unique integrability of the center implies unique integrability of both the center-stable in dimension 3. – rpotrie Mar 30 2012 at 14:24
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