arXiv subjects

Siming Tu

Publications and source records attributed to Siming Tu.

6 recordsLinked to original sources

arXiv2022

Sequence entropy tuples and mean sensitive tuples

Using the idea of local entropy theory, we characterize the sequence entropy tuple via mean forms of the sensitive tuple in both topological and measure-theoretical senses. For the measure-theoretical sense, we show that for an ergodic measure-preserving system, the $\mu$-sequence entropy tuple, the $\mu$-mean sensitive tuple and the $\mu$-sensitive in the mean tuple coincide, and give an example to show that the ergodicity condition is necessary. For the topological sense, we show that for a certain class of minimal systems, the mean sensitive tuple is the sequence entropy tuple.

math.DS↗

arXiv2021

Mean dimension of Bernstein spaces and universal real flows

We study the action of translation on the spaces of uniformly bounded continuous functions on the real line which are uniformly band-limited in a compact interval. We prove that two intervals themselves will decide if two spaces are topologically conjugate, while the length of an interval tells the mean dimension of a space. We also investigate universal real flows. We construct a sequence of compact invariant subsets of a space consisting of uniformly bounded smooth one-Lipschitz functions on the real line, which have mean dimension equal to one, such that all real flows can be equivariantly embedded in the translation on their product space. Moreover, we show that the countable self-product of any among them does not satisfy such a universal property. This, on the one hand, presents a more reasonable choice of a universal real flow with a view towards mean dimension, and on the other hand, clarifies a seemingly plausible impression; meanwhile, it refines the previously known results in this direction. Our proof goes through an approach of harmonic analysis. Furthermore, both the universal space that we provide and an embedding mapping which we build for any real flow are explicit.

math.DS↗

arXiv2018

A new universal real flow of the Hilbert-cubical type

We provide a new universal real flow of the Hilbert-cubical type. We prove that any real flow can be equivariantly embedded in the translation on $L(\mathbb{R})^\mathbb{N}$, where $L(\mathbb{R})$ denotes the space of $1$-Lipschitz functions $f:\mathbb{R}\to[0,1]$. Furthermore, all those functions in $L(\mathbb{R})^\mathbb{N}$ that are images of such embeddings can be chosen as $C^1$-functions.

math.DS↗

arXiv2016

Devaney chaos plus shadowing implies distributional chaos

We explore connections among the regional proximal relation, the asymptotic relation and the distal relation for a topological dynamical system with the shadowing property, and show that if a Devaney chaotic system has the shadowing property then it is distributionally chaotic.

math.DS↗

arXiv2013

Mean equicontinuity and mean sensitivity

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the later property must have zero entropy.

math.DS↗

arXiv2013

On proximality with Banach density one

Let $(X,T)$ be a topological dynamical system. A pair of points $(x,y)\in X^2$ is called Banach proximal if for any $\epsilon>0$, the set $\{n\in\mathbb{Z}:\ d(T^nx,T^ny)<\epsilon\}$ has Banach density one. We study the structure of the Banach proximal relation. An useful tool is the notion of the support of a topological dynamical system. We show that a dynamical system is strongly proximal if and only if every pair in $X^2$ is Banach proximal. A subset $S$ of $X$ is Banach scrambled if every two distinct points in $S$ form a Banach proximal pair but not asymptotic. We construct a dynamical system with the whole space being a Banach scrambled set. Even though the Banach proximal relation of the full shift is of first category, it has a dense Mycielski invariant Banach scrambled set. We also show that for an interval map it is Li-Yorke chaotic if and only if it has a Cantor Banach scrambled set.

math.DS↗