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Register a new userA note on mean equicontinuity
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In this note, it is shown that several results concerning mean equicontinuity proved before for minimal systems are actually held for general topological dynamical systems. Particularly, it turns out that a dynamical system is mean equicontinuous if and only if it is equicontinuous in the mean if and only if it is Banach (or Weyl) mean equicontinuous if and only if its regionally proximal relation is equal to the Banach proximal relation. Meanwhile, a relation is introduced such that the smallest closed invariant equivalence relation containing this relation induces the maximal mean equicontinuous factor for any system.
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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 weak mean equicontinuous properties for a countable discrete amenable group $G$ acting continuously on a compact metrizable space $X$ are studied. It is shown that the weak mean equicontinuity of $(X times X,G)$ is equivalent to the mean equicontinuity of $(X,G)$. Moreover, when $(X,G)$ has full measure center or $G$ is abelian, it is shown that $(X,G)$ is weak mean equicontinuous if and only if all points in $X$ are uniquely ergodic points and the map $x to mu_x^G$ is continuous, where $mu_x^G$ is the unique ergodic measure on ${ol{Orb(x)}, G}$.
We show that a homeomorphism of a semi-locally connected compact metric space is equicontinuous if and only if the distance between the iterates of a given point and a given subcontinuum (not containing that point) is bounded away from zero. This is false for general compact metric spaces. Moreover, homeomorphisms for which the conclusion of this result holds satisfy that the set of automorphic points contains those points where the space is not semi-locally connected.
We study dynamical systems that have bounded complexity with respect to three kinds of directional metrics: the directional Bowen metric $d_k^{vec{v},b}$, the directional max-mean metric $hat{d}_k^{vec{v},b}$ and the directional mean metric $overline{d}_k^{vec{v},b}$. It is shown that a $mathbb{Z}^q$-topological system $(X,T)$ has bounded topological complexity with respect to ${d_k^{vec{v},b}}$ (respectively ${hat{d}_k^{vec{v},b}}$) if and only if $T$ is $(vec{v},b)$-equicontinuous (respectively $(vec{v},b)$-equicontinuous in the mean). It turns out that a measure $mu$ has bounded complexity with respect to ${d_k^{vec{v},b}}$ if and only if $T$ is $(mu,vec{v},b)$-equicontinuous. Whats more, it is shown that $mu$ has bounded complexity with respect to ${overline{d}_k^{vec{v},b}}$ if and only if $mu$ has bounded complexity with respect to ${hat{d}_k^{vec{v},b}}$ if and only if $T$ is $(mu,vec{v},b)$-mean equicontinuous if and only if $T$ is $(mu,vec{v},b)$-equicontinuous in the mean if and only if $mu$ has $vec{v}$-discrete spectrum.
Let $ G $ be a discrete subgroup of PU(1,n). Then $ G $ acts on $mathbb {P}^n_mathbb C$ preserving the unit ball $mathbb {H}^n_mathbb {C}$, where it acts by isometries with respect to the Bergman metric. In this work we determine the equicontinuty region $Eq(G)$ of $G$ in $mathbb P^n_{mathbb C}$: It is the complement of the union of all complex projective hyperplanes in $mathbb {P}^n_{mathbb C}$ which are tangent to $partial mathbb {H}^n_mathbb {C}$ at points in the Chen-Greenberg limit set $Lambda_{CG}(G )$, a closed $G$-invariant subset of $partial mathbb {H}^n_mathbb {C}$, which is minimal for non-elementary groups. We also prove that the action on $Eq(G)$ is discontinuous.
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