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Kairan Liu

Publications and source records attributed to Kairan Liu.

10 recordsLinked to original sources
arXiv2025

Relative topological entropy and relative mean dimension of induced factors

We study the relation of relative topological entropy and relative mean dimension between a factor map and its induced factor map for amenable group actions. On the one hand, we prove that a factor map has zero relative topological entropy if and only if so does the induced factor map. On the other hand, we prove that a factor map has positive relative topological entropy if and only if the induced factor map has infinite relative mean dimension.

math.DS↗
arXiv2025

Local entropy theory, combinatorics, and local theory of Banach spaces

Each continuous action of a countably infinite discrete group $Γ$ on a compact metrizable space X induces a continuous action of $Γ$ on the space M(X) of Borel probability measures on X. We compare the local entropy theory for these two actions, and describe the relation between their IE-tuples. Several other types of tuples are also studied. Our main tool is a new combinatorial lemma. We also give an application of the combinatorial lemma to the local theory of Banach spaces.

math.DS↗
arXiv2023

Relative uniformly positive entropy of induced amenable group actions

Let $G$ be a countable infinite discrete amenable group.It should be noted that a $G$-system $(X,G)$ naturally induces a $G$-system $(\mathcal{M}(X),G)$, where $\mathcal{M}(X)$ denotes the space of Borel probability measures on the compact metric space $X$ endowed with the weak*-topology. A factor map $π\colon (X,G)\to(Y,G)$ between two $G$-systems induces a factor map $\widetildeπ\colon(\mathcal{M}(X),G)\to(\mathcal{M}(Y),G)$. It turns out that $\widetildeπ$ is open if and only if $π$ is open. When $Y$ is fully supported, it is shown that $π$ has relative uniformly positive entropy if and only if $\widetildeπ$ has relative uniformly positive entropy.

math.DS↗
arXiv2020

Mean Li-Yorke chaos along polynomials of several variables and prime numbers

In this paper, for any given polynomial, by analyzing the limiting behavior of ergodic averages along polynomials of several variables and prime numbers, we prove that for a topology dynamical system, positive entropy implies mean Li-Yoke chaos along non-constant polynomials of several variables and prime numbers.

math.DS↗
arXiv2020

Time-restricted sensitivity and entropy

In this paper, we consider measure-theoretical restricted sensitivity and topological restricted sensitivities by restricting the first sensitive time. For a given topological dynamical system, we define measure-theoretical restricted asymptotic rate with respect to sensitivity, and obtain that it equal to the reciprocal of the Brin-Katok local entropy for almost every point. For topological version we have similar definitions and conclusions.

math.DS↗
arXiv2020

$Δ$-weakly mixing subsets along a collection of sequences of integers

In this paper, we propose a mild condition, named Condition $(**)$, for collections of sequence of integers and show that for any measure preserving system the Pinsker $σ$-algebra is a characteristic $σ$-algebra for the averages along a collection satisfying Condition $(**)$. We introduce the notion of $Δ$-weakly mixing subsets along a collection of sequences of integers and show that positive topological entropy implies the existence of $Δ$-weakly mixing subsets along a collection of "good" sequences. As a consequence, we show that positive topological entropy implies multi-variant Li-Yorke chaos along polynomial times of the shift prime numbers.

math.DS↗
arXiv2019

Topological entropy of nonautonomous dynamical systems

Let $\mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^\ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system $(X,\{f_n\}_{n=1}^{+\infty})$ vanishes, then so does that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$; moreover, once the topological entropy of $(X,\{f_n\}_{n=1}^{+\infty})$ is positive, that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$ jumps to infinity. In contrast to Bowen's inequality, we construct a nonautonomous dynamical system whose topological entropy is not preserved under a finite-to-one extension.

math.DS↗
arXiv2019

$Δ$-Weakly Mixing Subset In Positive Entropy Actions Of A Nilpotent Group

The notion of $Δ$-weakly mixing subsets is introduced for countable torsion-free discrete group actions. It is shown that for a finitely generated torsion-free discrete nilpotent group action, positive topological entropy implies the existence of $Δ$-weakly mixing subsets, and while there exists a finitely generated torsion-free discrete solvable group action which has positive topological entropy but without any $Δ$-weakly mixing subsets.

math.DS↗
arXiv2019

Auslander-Yorkes type dichotomy theorems for stronger version r-sensitivity

In this paper, for r in N with r>=2 we consider several stronger version r-sensitivities and measure-theoretical r-sensitivities by analysing subsets of nonnegative integers, for which the r-sensitivity occurs. We obtain an Auslander-Yorke's type dichotomy theorem: a minimal topological dynamical system is either thickly r-sensitive or an almost m to one extension of its maximal equicontinuousfactor for some m in {1,2,...,r-1}.

math.DS↗