Related papers: Sofic mean dimension of typical actions and a comp…

In previous work, I introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new…

Let $G$ be a finite group and $\phi\colon V\to W$ an equivariant morphism of finite dimensional $G$-modules. We say that $\phi$ is faithful if $G$ acts faithfully on $\phi(V)$. The covariant dimension of $G$ is the minimum of the dimension…

We continue our study of when topological and measure-theoretic entropy agree for algebraic action of sofic groups. Specifically, we provide a new abstract method to prove that an algebraic action is strongly sofic. The method is based on…

Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given…

We construct p.m.p. group actions that are not local-global limits of sequences of finite graphs. Moreover, they do not weakly contain any sequence of finite labeled graphs. Our methods are based on the study of almost automorphisms of…

For a locally compact sofic group continuously acting on a compact metric space, we first study the relative sofic entropy and prove an additive inequality relating sofic entropy and relative sofic entropy. Moreover, it is shown that the…

We introduce the notion of dynamic asymptotic dimension growth for actions of discrete groups on compact spaces, and more generally for locally compact \'etale groupoids. Using the work of Bartels, L\"uck, and Reich, we bridge asymptotic…

In this paper we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire's sense) invariant measure has, for each $q>0$, zero lower $q$-generalized fractal dimension.…

Given a two-sided shift space on a finite alphabet and a continuous potential function, we give conditions under which an equilibrium measure can be described using a construction analogous to Hausdorff measure that goes back to the work of…

We define an infinite graded graph of ordered pairs and a~canonical action of the group $\mathbb{Z}$ (the adic action) and of the infinite sum of groups of order two~$\mathcal{D}=\sum_1^{\infty} \mathbb{Z}/2\mathbb{Z}$ on the path space of…

In this work, we prove that every SFT, sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every…

Let $X$ be a full-shift on the alphabet $[0, 1]^a$ and let $(Y, S)$ be an arbitrary dynamical system. We prove that any equivariant continuous map from $X$ to $Y$ has conditional metric mean dimension not less than $a-\mathrm{mdim}(Y, S)$.…

In this article we develop a notion of soficity for actions of countable groups on sets. We show two equivalent perspectives, several natural properties and examples. Notable examples include arbitrary actions of both amenable groups and…

We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, effectively showing how every hyperlinear approximation to such a group is simulated by a suitable sofic approximation. The…

A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag--Solitar group, is a…

In the context of (not necessarily minimal) actions, we consider the mean diameter and use it to characterize regular factor maps. Building on this characterization, we prove that an action is diam-mean equicontinuous if and only if it is a…

We study directional mean dimension of $\mathbb{Z}^k$-actions (where $k$ is a positive integer). On the one hand, we show that there is a $\mathbb{Z}^2$-action whose directional mean dimension (considered as a $[0,+\infty]$-valued function…

We show that the class of sofic actions is closed under direct products and contains a (non-unique) maximal element in the weak containment order. For any sofic group we construct nice sofic approximations such that all the sofic actions…

We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $\Gamma$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced…

The Hermite-Hadamard inequality states that the average value of a convex function on an interval is bounded from above by the average value of the function at the endpoints of the interval. We provide a generalization to higher dimensions:…