Searcharxiv⌕ Search

arXiv subjects

Ruxi Shi

Publications and source records attributed to Ruxi Shi.

31 records · Page 2Linked to original sources
arXiv2021

Zero-dimensional and symbolic extensions of topological flows

A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [Bur19] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-$t$ map admits an extension by a subshift for any $t\neq 0$. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold more true for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on $\{0,1\}^{\mathbb Z}$ with a roof function $f$ vanishing at the zero sequence $0^\infty$ admits a principal symbolic extension or not depending on the smoothness of $f$ at $0^\infty$.

math.DS↗
arXiv2021

Divergent coindex sequence for dynamical systems

When a finite group freely acts on a topological space, we can define its index and coindex. They roughly measure the size of the given action. We explore the interaction between this index theory and topological dynamics. Given a fixed-point free dynamical system, the set of $p$-periodic points admits a natural free action of $\mathbb{Z}/p\mathbb{Z}$ for each prime number $p$. We are interested in the growth of its index and coindex as $p\to \infty$. Our main result shows that there exists a fixed-point free dynamical system having the divergent coindex sequence. This solves a problem posed by [TTY20].

math.DS↗
arXiv2021

Finite mean dimesnion and marker property

In this paper, we develop the theory of $\mathbb{Z}_p$-index which has been introduced by Tsukamoto, Tsutaya and Yoshinaga. As an application, we show that given any positive number, there exists a dynamical system with mean dimension equal to such number such that it does not have the marker property.

math.DS↗
arXiv2021

On variational principles for metric mean dimension

In this note, we show several variational principles for metric mean dimension. First we prove a variational principles in terms of Shapira's entropy related to finite open covers. Second we establish a variational principle in terms of Katok's entropy. Finally using these two variational principles we develop a variational principle in terms of Brin-Katok local entropy.

math.DS↗
arXiv2020

Embedding theorems for discrete dynamical systems and topological flows

In this paper, we investigate the embeddings for topological flows. We prove an embedding theorem for discrete topological system. Our results apply to suspension flows via constant function, and for this case we show an embedding theorem for suspension flows and give a new proof of Gutman-Jin embedding theorem.

math.DS↗
arXiv2020

On $p$-adic spectral measures

A Borel probability measure $μ$ on a locally compact group is called a spectral measure if there exists a subset of continuous group characters which forms an orthogonal basis of the Hilbert space $L^2(μ)$. In this paper, we characterize all spectral measures in the field $\mathbb{Q}_p$ of $p$-adic numbers.

math.FA↗
arXiv2019

Fuglede's conjecture holds on cyclic groups $\mathbb{Z}_{pqr}$

Fuglede's spectral set conjecture states that a subset $Ω$ of a locally compact abelian group $G$ tiles the group by translation if and only if there exists a subset of continuous group characters which is an orthogonal basis of $L^2(Ω)$. We prove that Fuglede's conjecture holds on cyclic groups $\mathbb{Z}_{pqr}$ with $p,q,r$ distinct primes.

math.CA↗
arXiv2019

Equi-distributed property and spectral set conjecture on $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$

In this paper, we show an equi-disctributed property in $2$-dimensional finite abelian groups $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ where $p$ is a prime number. By using this equi-disctributed property, we prove that Fuglede's spectral set conjecture holds on groups $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$, namely, a set in $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ is a spectral set if and only if it is a tile.

math.CA↗
arXiv2018

Construction of some Chowla sequences

For numerical sequences taking values $0$ or complex numbers of modulus $1$, we define Chowla property and Sarnak property. We prove that Chowla property implies Sarnak property. We also prove that for Lebesgue almost every $β>1$, the sequence $(e^{2πβ^n})_{n\in \mathbb{N}}$ shares Chowla property and consequently is orthogonal to all topological dynamical systems of zero entropy. It is also discussed whether the samples of a given random sequence have Chowla property almost surely. Some dependent random sequences having almost surely Chowla property are constructed.

math.DS↗
arXiv2018

Equivalent definitions of oscillating sequences of higher orders

An oscillating sequence of order $d$ is defined by the linearly disjointness from all $\{e^{2πi P(n)} \}_{n=1}^{\infty}$ for all real polynomials $P$ of degree smaller or equal to $d$. A fully oscillating sequence is defined to be an oscillating sequence of all orders. In this paper, we give several equivalent definitions of such sequences in terms of their disjointness from different dynamical systems on tori.

math.DS↗
arXiv2016

Compact Open Spectral Sets In $\mathbb{Q}_p$

In this article, we prove that a compact open set in the field $\mathbb{Q}_p$ of $p$-adic numbers is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation, and also if and only if it is $p$-homogeneous which is easy to check. We also characterize spectral sets in $\mathbb{Z}/p^n \mathbb{Z}$ ($p\ge 2$ prime, $n\ge 1$ integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in $\mathbb{Q}_p$, some of which are self-similar measures.

math.FA↗
arXiv2015

Fuglede's conjecture holds in $\mathbb{Q}_p$

Fuglede's conjecture in $\mathbb{Q}_p$ is proved. That is to say, a Borel set of positive and finite Haar measure in $\mathbb{Q}_p$ is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation.

math.CA↗