Fractal entropy of nonautonomous systems

Abstract

We define formulas of entropy dimension for a nonautonomous dynamical system consisting of a sequence of continuous self-maps of a compact metric space. This study reveals analogues of basic propositions for entropy dimension, such as the power rule, product rule and commutativity, etc. These properties allow us to convert to an equality an inequality found by de Carvalho (1997) concerning the product rule for the autonomous dynamical system. We also prove a subadditivity rule of entropy dimension for one-dimensional dynamics based on our previous work.

Topological pressure dimension for almost additive potentials

“…Moreover, various authors have introduced several refinements of entropy, including slow entropy [30], measure-theoretic complexity [25], and entropy-like invariants for noninvertible maps [29,39,19]. The authors [18,36,37,34] extended and studied as above some entropy-like invariants for the non-autonomous discrete dynamical systems given by a sequence of continuous self-maps of a compact topological space.…”

Preimage entropy dimension of topological dynamical systems

“…Several important pre-image entropy invariant, such as pointwise pre-image, pointwise branch entropy, partial pre-image entropy, and bundle-like pre-image entropy, etc., have been introduced and their relationships with topological entropy have been established. The authors [9,20,21,19] extended and studied as above some entropy-like invariants for the non-autonomous discrete dynamical systems given by a sequence of continuous self-maps of a compact topological space.…”