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Register a new userAffine transformation with zero entropy and nilsystems
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Zhengxing Lian
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In this paper, we study affine transformations on tori, nilmanifolds and compact abelian groups. For these systems, we show that an equivalent condition for zero entropy is the orbit closure of each point has a nice structure. To be precise, the affine systems on those spaces are zero entropy if and only if the orbit closure of each point is isomorphic to an inverse limit of nilsystems.
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We prove that the maximal infinite step pro-nilfactor $X_infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $pi:X rightarrow X_infty$, the induced open extension $pi^*:X^* rightarrow X^*_infty$ has the following property: for $x$ in a dense $G_delta$ set of $X^*$, the orbit closure $L_x=overline{{mathcal{O}}}((x,x,ldots,x), Ttimes T^2times ldots times T^d)$ is $(pi^*)^{(d)}$-saturated, i.e. $L_x=((pi^*)^{(d)})^{-1}(pi^*)^{(d)}(L_x)$. Using results derived from the above fact, we are able to answer several open questions: (1) if $(X,T^k)$ is minimal for some $kge 2$, then for any $din {mathbb N}$ and any $0le j<k$ there is a sequence ${n_i}$ of $mathbb Z$ with $n_iequiv j (text{mod} k)$ such that $T^{n_i}xrightarrow x, T^{2n_i}xrightarrow x, ldots, T^{dn_i}xrightarrow x$ for $x$ in a dense $G_delta$ subset of $X$; (2) if $(X,T)$ is totally minimal, then ${T^{n^2}x:nin {mathbb Z}}$ is dense in $X$ for $x$ in a dense $G_delta$ subset of $X$; (3) for any $dinmathbb N$ and any minimal system, which is an open extension of its maximal distal factor, ${bf RP}^{[d]}={bf AP}^{[d]}$, where the latter is the regionally proximal relation of order $d$ along arithmetic progressions.
In this paper, we study dynamics of maps on quasi-graphs characterizing their invariant measures. In particular, we prove that every invariant measure of quasi-graph map with zero topological entropy has discrete spectrum. Additionally, we obtain an analog of Llibre-Misiurewiczs result relating positive topological entropy with existence of topological horseshoes. We also study dynamics on dendrites and show that if a continuous map on a dendrite, whose set of all endpoints is closed and has only finitely many accumulation points, has zero topological entropy, then every invariant measure supported on an orbit closure has discrete spectrum.
In this paper, it is shown that for $dinmathbb{N}$, a minimal system $(X,T)$ is a $d$-step pro-nilsystem if its enveloping semigroup is a $d$-step top-nilpotent group, answering an open question by Donoso. Thus, combining the previous result of Donoso, it turns out that a minimal system $(X,T)$ is a $d$-step pro-nilsystem if and only if its enveloping semigroup is a $d$-step top-nilpotent group.
The regionally proximal relation of order $d$ along arithmetic progressions, namely ${bf AP}^{[d]}$ for $din N$, is introduced and investigated. It turns out that if $(X,T)$ is a topological dynamical system with ${bf AP}^{[d]}=Delta$, then each ergodic measure of $(X,T)$ is isomorphic to a $d$-step pro-nilsystem, and thus $(X,T)$ has zero entropy. Moreover, it is shown that if $(X,T)$ is a strictly ergodic distal system with the property that the maximal topological and measurable $d$-step pro-nilsystems are isomorphic, then ${bf AP}^{[d]}={bf RP}^{[d]}$ for each $din {mathbb N}$. It follows that for a minimal $infty$-pro-nilsystem, ${bf AP}^{[d]}={bf RP}^{[d]}$ for each $din {mathbb N}$. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.
We analyze some properties of maximizing stationary Markov probabilities on the Bernoulli space $[0,1]^mathbb{N}$, More precisely, we consider ergodic optimization for a continuous potential $A$, where $A: [0,1]^mathbb{N}to mathbb{R}$ which depends only on the two first coordinates. We are interested in finding stationary Markov probabilities $mu_infty$ on $ [0,1]^mathbb{N}$ that maximize the value $ int A d mu,$ among all stationary Markov probabilities $mu$ on $[0,1]^mathbb{N}$. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities $mu_beta$ which weakly converges to $mu_infty$. The probabilities $mu_beta$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure. Under the hypothesis of $A$ being $C^2$ and the twist condition, that is, $frac{partial^2 A}{partial_x partial_y} (x,y) eq 0$, for all $(x,y) in [0,1]^2$, we show the graph property.
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