Related papers: Set-theoretical entropies of weighted generalized …

In the following text for arbitrary $X$ with at least two elements, nonempty set $\Gamma$ and self-map $\varphi:\Gamma\to\Gamma$ we prove the set-theoretical entropy of generalized shift $\sigma_\varphi:X^\Gamma\to X^\Gamma$…

In the following text for vector space $V$ over field $F$ we compute all eigenvalues of weighted generalized shift $\sigma_{\varphi,\mathfrak{w}}:V^\Gamma\to V^\Gamma$ (and generalized shift $\sigma_\varphi:V^\Gamma\to V^\Gamma$) for…

For a set $\Gamma$, a function $\lambda:\Gamma\to \Gamma$ and a non-trivial abelian group $K$, the generalized shift $\sigma_\lambda:K^\Gamma\to K^\Gamma$ is defined by $(x_i)_{i\in \Gamma}\mapsto (x_{\lambda(i)})_{i\in\Gamma}$. In this…

In the following text we prove that for finite discrete $X$ with at least two elements and infinite $\Gamma$, the generalized shift transformation semigroup $({\mathcal S},X^\Gamma)$ is equicontinuous (resp. has at least an equicontinuous…

For every finite-to-one map $\lambda:\Gamma\to\Gamma$ and for every abelian group $K$, the generalized shift $\sigma_\lambda$ of the direct sum $\bigoplus_\Gamma K$ is the endomorphism defined by…

Let $\Gamma $ be an infinite discrete group and $\mathsf{A}\subset \Gamma $ a nonempty finite subset. The set of permutations $\sigma $ of $\Gamma $ such that $s^{-1}\sigma (s)\in \mathsf{A}$ for every $s\in \Gamma $ can be identified with…

Suppose $X$ is a finite discrete space with at least two elements, $\Gamma$ is a nonempty countable set, and consider self--map $\varphi:\Gamma\to\Gamma$. We prove that the generalized shift $\sigma_\varphi:X^\Gamma\to X^\Gamma$ with…

For a continuous map $f$ on a compact metric space we study the geometry and entropy of the generalized rotation set $\R(\Phi)$. Here $\Phi=(\phi_1,...,\phi_m)$ is a $m$-dimensional continuous potential and $\R(\Phi)$ is the set of all…

For a finite discrete topological space $X$ with at least two elements, a nonempty set $\Gamma$, and a map $\varphi:\Gamma\to\Gamma$, $\sigma_\varphi:X^\Gamma\to X^\Gamma$ with $\sigma_\varphi((x_\alpha)_{\alpha\in\Gamma})=…

We partially generalize Peters' formula on modules over the group ring ${\mathbb F} \Gamma$ for a given finite field ${\mathbb F}$ and a sofic group $\Gamma$. It is also discussed that how the values of entropy are related to the zero…

Let $\varphi\colon\Gamma\to G$ be a homomorphism of groups. In this paper we introduce the notion of a subnormal map (the inclusion of a subnormal subgroup into a group being a basic prototype). We then consider factorizations…

The $S$-gap shifts have a dynamically and combinatorially rich structure. Dynamical properties of the $S$-gap shift can be related to the properties of the set $S$. This interplay is particularly interesting when $S$ is not syndetic such as…

$S$-gap shifts are a well-studied class of shift spaces, which has led to several proposed generalizations. This paper introduces a new class of shift spaces called $\mathcal{S}$-graph shifts whose essential structure is encoded in a novel…

Let $G$ be a finitely generated amenable group. We study the space of shifts on $G$ over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that more generally the shifts of entropy $c$ are…

We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any…

In this work we consider an ensemble of random $\mathbb{Z}^d$-shifts of finite type ($\mathbb{Z}^d$-SFTs) and prove several results concerning the behavior of typical systems with respect to emptiness, entropy, and periodic points. These…

Let $G=\left\langle S|R_{A}\right\rangle $ be a semigroup with generating set $ S$ and equivalences $R_{A}$ among $S$ determined by a matrix $A$. This paper investigates the complexity of $G$-shift spaces by yielding the topological…

The required absence of global symmetries in quantum gravity has been used to imply that all non-compact gauge theories are in the swampland. This argument stems from the idea that non-compact gauge symmetries always seem to be accompanied…

In the following text for $p\in[1,\infty]$, nonzero cardinal number $\tau$, self--map $\varphi:\tau\to\tau$ if there exists $N\in\mathbb{N}$ such that $\varphi^{-1}(\alpha)$ has at most $N$ elements for each $\alpha<\tau$, and operators…

The first part of the paper centers in the study of embeddability between partially commutative groups. In [KK], for a finite simplicial graph $\Gamma$, the authors introduce an infinite, locally infinite graph $\Gamma^e$, called the…