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Entropy is a very useful concept from physics that tries to explain how a system behaves from a point of view of the thermodynamics. However, there are two ways to explain entropy, and it depends on if we are studying a microsystem or a…
We study the topological complexities of relative entropy zero extensions acted by countableinfinite amenable groups. Firstly, for a given Folner sequence $\{F_n\}_{n=0}^\infty$, we define respectively the relative entropy dimensions and…
The entropy production is commonly interpreted as measuring the distance from equilibrium. However, this explanation lacks a rigorous description due to the absence of a natural equilibrium measure. The present analysis formalizes this…
We develop in this paper some general techniques to analyze action sets of small doubling for probability measure-preserving actions of amenable groups. As an application of these techniques, we prove a dynamical generalization of Kneser's…
We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to…
The characterization of irreversibility in general quantum processes is an open problem of increasing techno- logical relevance. Yet, the tools currently available to this aim are mostly limited to the assessment of dynamics induced by…
In this paper, we show that there is a net for amenable transformation groups like F{\o}lner net for amenable groups and investigate amenability of a transformation group constructed by semidirect product of groups. We introduce inner…
Entropy production is the crucial quantity characterizing irreversible phenomena and the second law of thermodynamics. Yet, a ubiquitous definition eludes consensus. Given that entropy production arises from incomplete access to…
In this paper we introduce the definition of entropy for a partial $\mathbb{Z}$-action. We show that the definition of partial entropy is an extension of the definition of topological entropy for a $\mathbb Z$-action. We also prove that the…
Different notions of entropy play a fundamental role in the classical theory of dynamical systems. Unlike many other concepts used to analyze autonomous dynamics, both measure-theoretic and topological entropy can be extended quite…
We introduce the notions of definable amenability and extreme definable amenability for groups in continuous structures and conduct an extensive analysis of them, drawing parallels with the classical first-order case. We characterize both…
For any discrete time dynamical system with a rational evolution, we define an entropy, which is a global index of complexity for the evolution map. We analyze its basic properties and its relations to the singularities and the…
The scaling entropy of a p.m.p. action is a slow-entropy type invariant that characterizes the intermediate growth of entropy in a dynamical system. An amenable group $G$ has a scaling entropy growth gap if the scaling entropy of any its…
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…
Generalized entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information…
Given the full shift over a countable state space on a countable amenable group, we develop its thermodynamic formalism. First, we introduce the concept of pressure and, using tiling techniques, prove its existence and further properties…
We introduce the notion of a contractible subshift. This is a strengthening of the notion of strong irreducibility, where we require that the gluings are given by a block map. We show that a subshift is a retract of a full shift if and only…
We develop a semigroup approach to representation theory for pro-Lie groups satisfying suitable amenability conditions. As an application of our approach, we establish a one-to-one correspondence between equivalence classes of unitary…
We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. On route, we investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the…
We generalize the notion of isoperimetric profiles of finitely generated groups to their actions by measuring the boundary of finite subgraphings of the orbit graphing. We prove that like the classical isoperimetric profiles for groups,…