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It is certainly the case that classifying the minimal subsystems of homeomorphisms of compact 2-manifolds presents profound and fundamental difficulties. This is because some very simple transformations, such as analytic diffeomorphisms of the 2-torus, have extremely rich families of minimal sets.

Let $T \colon X \to X$ be a linear Anosov diffeomorphism of the 2-torus. The topological entropy of $T$ is finite and positive but may be arbitrarily large. When this entropy If a natural number $k$ is large enoughspecified, then we may find a linear Anosov diffeomorphism $T$ of the 2-torus $X$ such that the shift transformation on $k$ symbols (for any previously-specified natural number $k$) may be homeomorphically embedded into the dynamical system $(X,T)$ as a compact invariant subset. In particular, every minimal subsystem of the $k$-shift embeds into $(X,T)$ as a minimal subsystem.

This is problematic because the $k$-shift has an enormous number of minimal subsystems, all of which will be inherited by the Anosov system. Indeed, the combinatorial version of the Jewett-Krieger theorem implies that every ergodic measure-preserving transformation of an abstract probability space which has entropy strictly less than $\log k$ may be embedded into the $k$-shift as a uniquely ergodic minimal subsystem. In particular, for a linear Anosov diffeomorphism of the 2-torus with topological entropy large enough, every ergodic measurable dynamical system with measure-theoretic entropy up to some threshold arises as a uniquely ergodic minimal subsystem.

This already presents us with an enormous number of minimal subsystems, because for each $h \geq 0$ there exist uncountably many ergodic measurable dynamical systems of entropy $h$ which are not pairwise equivalent. This is then compounded by the fact that some minimal systems of $(X,T)$ will not arise from such an embedding, the fact that the embedding of the abstract ergodic system into the $k$-shift is in general not unique, the fact that the embedding of the $k$-shift into $(X,T)$ is in general not unique, and the fact that the $k$-shift itself has additional minimal subsystems. Indeed, there is a further theorem due to Denker, Grillenberger and Sigmund which implies that for any finite collection of abstract ergodic transformations all having entropy strictly below $\log k$, we can find a minimal subsystem of the $k$-shift which has an embedded copy of each of these transformations as its only ergodic measures.

On the basis of the above considerations I think that a satisfactory classification of the minimal subsystems of homeomorphisms of the 2-torus is improbable!

2 edited body

It is certainly the case that classifying the minimal subsystems of homeomorphisms of compact 2-manifolds presents profound and fundamental difficulties. This is because some very simple transformations, such as analytic diffeomorphisms of the 2-torus, have extremely rich families of minimal sets.

Let $T \colon X \to X$ be a linear Anosov diffeomorphism of the 2-torus. The topological entropy of $T$ is finite and positive but may be arbitrarily large. When this entropy is large enough, the shift transformation on $k$ symbols (for any previously-specified natural number $k$) may be homeomorphically embedded into the dynamical system $(X,T)$ as a compact invariant subset. In particular, every minimal subsystem of the $k$-shift embeds into $(X,T)$ as a minimal subsystem.

This is problematic because the $k$-shift has an enormous number of minimal subsystems, all of which will be inherited by the Anosov system. Indeed, the combinatorial version of the Jewett-Krieger theorem implies that every ergodic measure-preserving transformation of an abstract probability space which has entropy strictly less than $\log k$ may be embedded into the $k$-shift as a uniquely ergodic minimal subsystem. In particular, for a linear Anosov diffeomorphism of the 2-torus with topological entropy large enough, every ergodic measurable dynamical system with measure-theoretic entropy up to some threshold arises as a uniquely ergodic minimal subsystem.

This already presents us with an enormous number of minimal subsystems, because for each $h \geq 0$ there exist uncountably many ergodic measurable dynamical systems of entropy $h$ which are not pairwise equivalent. This is then compounded by the fact that some minimal systems of $(X,T)$ will not arise from such an embedding, the fact that the embedding of the abstract ergodic system into the $k$-shift is in general nor not unique, the fact that the embedding of the $k$-shift into $(X,T)$ is not in general not unique, and the fact that the $k$-shift itself has additional minimal subsystems. Indeed, there is a further theorem due to Denker, Grillenberger and Sigmund which implies that for any finite collection of abstract ergodic transformations all having entropy strictly below $\log k$, we can find a minimal subsystem of the $k$-shift which has an embedded copy of each of these transformations as its only ergodic measures.

On the basis of the above considerations I think that a satisfactory classification of the minimal subsystems of homeomorphisms of the 2-torus is improbable!

1

It is certainly the case that classifying the minimal subsystems of homeomorphisms of compact 2-manifolds presents profound and fundamental difficulties. This is because some very simple transformations, such as analytic diffeomorphisms of the 2-torus, have extremely rich families of minimal sets.

Let $T \colon X \to X$ be a linear Anosov diffeomorphism of the 2-torus. The topological entropy of $T$ is finite and positive but may be arbitrarily large. When this entropy is large enough, the shift transformation on $k$ symbols (for any previously-specified natural number $k$) may be homeomorphically embedded into the dynamical system $(X,T)$ as a compact invariant subset. In particular, every minimal subsystem of the $k$-shift embeds into $(X,T)$ as a minimal subsystem.

This is problematic because the $k$-shift has an enormous number of minimal subsystems, all of which will be inherited by the Anosov system. Indeed, the combinatorial version of the Jewett-Krieger theorem implies that every ergodic measure-preserving transformation of an abstract probability space which has entropy strictly less than $\log k$ may be embedded into the $k$-shift as a uniquely ergodic minimal subsystem. In particular, for a linear Anosov diffeomorphism of the 2-torus with topological entropy large enough, every ergodic measurable dynamical system with measure-theoretic entropy up to some threshold arises as a uniquely ergodic minimal subsystem.

This already presents us with an enormous number of minimal subsystems, because for each $h \geq 0$ there exist uncountably many ergodic measurable dynamical systems of entropy $h$ which are not pairwise equivalent. This is then compounded by the fact that some minimal systems of $(X,T)$ will not arise from such an embedding, the fact that the embedding of the abstract ergodic system into the $k$-shift is in general nor unique, the fact that the embedding of the $k$-shift into $(X,T)$ is not in general unique, and the fact that the $k$-shift itself has additional minimal subsystems. Indeed, there is a further theorem due to Denker, Grillenberger and Sigmund which implies that for any finite collection of abstract ergodic transformations all having entropy strictly below $\log k$, we can find a minimal subsystem of the $k$-shift which has an embedded copy of each of these transformations as its only ergodic measures.

On the basis of the above considerations I think that a satisfactory classification of the minimal subsystems of homeomorphisms of the 2-torus is improbable!
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