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Register a new userSensitivity, proximal extension and higher order almost automorphy
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Let $(X,T)$ be a topological dynamical system, and $mathcal{F}$ be a family of subsets of $mathbb{Z}_+$. $(X,T)$ is strongly $mathcal{F}$-sensitive, if there is $delta>0$ such that for each non-empty open subset $U$, there are $x,yin U$ with ${ninmathbb{Z}_+: d(T^nx,T^ny)>delta}inmathcal{F}$. Let $mathcal{F}_t$ (resp. $mathcal{F}_{ip}$, $mathcal{F}_{fip}$) be consisting of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets). The following Auslander-Yorkes type dichotomy theorems are obtained: (1) a minimal system is either strongly $mathcal{F}_{fip}$-sensitive or an almost one-to-one extension of its $infty$-step nilfactor. (2) a minimal system is either strongly $mathcal{F}_{ip}$-sensitive or an almost one-to-one extension of its maximal distal factor. (3) a minimal system is either strongly $mathcal{F}_{t}$-sensitive or a proximal extension of its maximal distal factor.
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In this paper, it is shown that for a minimal system $(X,T)$ and $d,kin mathbb{N}$, if $(x,x_i)$ is regionally proximal of order $d$ for $1leq ileq k$, then $(x,x_1,ldots,x_k)$ is $(k+1)$-regionally proximal of order $d$. Meanwhile, we introduce the notion of $mathrm{IN}^{[d]}$-pair: for a dynamical system $(X,T)$ and $din mathbb{N}$, a pair $(x_0,x_1)in Xtimes X$ is called an $mathrm{IN}^{[d]}$-pair if for any $kin mathbb{N}$ and any neighborhoods $U_0 ,U_1 $ of $x_0$ and $x_1$ respectively, there exist integers $p_j^{(i)},1leq ileq k,$ $1leq jleq d$ such that $$ bigcup_{i=1}^k{ p_1^{(i)}epsilon(1)+ldots+p_d^{(i)} epsilon(d):epsilon(j)in {0,1},1leq jleq d}backslash {0}subset mathrm{Ind}(U_0,U_1), $$ where $mathrm{Ind}(U_0,U_1)$ denotes the collection of all independence sets for $(U_0,U_1)$. It turns out that for a minimal system, if it dose not contain any nontrivial $mathrm{IN}^{[d]}$-pair, then it is an almost one-to-one extension of its maximal factor of order $d$.
The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of a minimal topological dynamical system has trivial (one point) fibres. In other words, we characterize when minimal mean equicontinuous systems are almost automorphic. Furthermore, we investigate another natural subclass of mean equicontinuous systems, so-called diam-mean equicontinuous systems, and show that a minimal system is diam-mean equicontinuous if and only if the maximal equicontinuous factor is regular (the points with trivial fibres have full Haar measure). Combined with previous results in the field, this provides a natural characterization for every step of a natural hierarchy for strictly ergodic topological models of ergodic systems with discrete spectrum. We also construct an example of a transitive almost diam-mean equicontinuous system with positive topological entropy, and we give a partial answer to a question of Furstenberg related to multiple recurrence.
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 introduce an infinite sequence of higher order Schwarzian derivatives closely related to the theory of monotone matrix functions. We generalize the classical Koebe lemma to maps with positive Schwarzian derivatives up to some order, obtaining control over derivatives of high order. For a large class of multimodal interval maps we show that all inverse branches of first return maps to sufficiently small neighbourhoods of critical values have their higher order Schwarzian derivatives positive up to any given order.
Let $X$ be a regular curve and $n$ be a positive integer such that for every nonempty open set $Usubset X$, there is a nonempty connected open set $Vsubset U$ with the cardinality $|partial_X(V)|leq n$. We show that if $X$ admits a sensitive action of a group $G$, then $G$ contains a free subsemigroup and the action has positive geometric entropy. As a corollary, $X$ admits no sensitive nilpotent group action.
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