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is there an example for a dynamical system $(X,T)$ where $X$ is a compact space and $T:X \to X$ is a continous map, s.t $\Pi(T) = \emptyset$, where $$\Pi(T) = \bigcup_{n=1}^{\infty} \Pi_n(T), \Pi_n(T) = \left\{ x \in X : T^n x =x \right\}$$

and its topological entropy, $h(T)>0$. http://en.wikipedia.org/wiki/Topological_entropy

My reasoning is that if $\Pi(T) = \emptyset$ then $\forall m\in \mathbb{N} : T^m x \neq x$. And from the definition of topological entropy that I want to calculate the metric $d_n(x,T^m x) = \max_{0\leq k \leq n-1} d(T^k x , T^{k+m} x)$; so I believe that if $T$ is not injective then I can find such a example where $h(T)>0$ and $\Pi(T)=\emptyset$.

But I am not sure how to define $X$ and the metric that goes alongside it. any hints references, are welcomed.

Julián Aguirre
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MathematicalPhysicist
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1 Answers1

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There is a classical example of a minimal homeomorphism $T$ (i.e. a continuous automorphism) with positive entropy provided by Mary Rees A minimal positive entropy homeomorphism of the 2 torus. J. London Math. Soc.23(1981), 537–550.

[A dynamical system $(X,T)$ is called minimal if $X$ does not contain any non-empty, proper, closed $T-$invariant subset. In such a case we also say that the map $T$ itself is minimal. See http://www.scholarpedia.org/article/Minimal_dynamical_systems for a nice discussion of minimal dynamical systems}. In particular, if $T$ is minimal and $X$ not discrete then $\Pi(T)=\emptyset$, but the property of being minmal is generally stronger than having empty periodic set (ebit)]

This theorem has been recently been generalised, see for example the paper by Beguin, S. Crovisier and F. Le Roux http://arxiv.org/pdf/math/0605438.pdf, who describe a general setting for what they call the Denjoy-Rees technique. This general setting includes as particular cases the construction of various “Denjoy counter-examples” in any dimension, and Rees construction of a minimal homeomorphism of $\mathbb T^d$ with positive topological entropy.

As the same paper points out, those results are in contrast to sufficinetly regular hyperbolic diffeomorphisms, which always have plenty of periodic orbits: a theorem of A. Katok (Lyapounov exponents, entropy and periodic orbits for diffeomorphisms. Pub Math IHES 51 (1980), 131–173) states that, if $T$ is a $C^{1+\alpha}$ diffeomorphism of a compact surface $X$ with positive topological entropy, then there exists an $T$-invariant compact set $\Lambda \subset X$ such that some power of $T|\Lambda$ is conjugate to a full shift. In particular, a $C^{1+\alpha}$ diffeomorphism of a compact surface with positive topological entropy cannot be minimal and will have $\Pi(T)\ne\emptyset$.

I'm not aware of any published results towards the best possible regularity (is it $C^1$ ?) of minimal automorphisms $T$ of $\mathbb T^2$ with positive entropy - any comments are welcome!

Raphael Albrecht
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