Entropy Formulas For Dynamical Systems With Mistakes

TL;DR

This paper investigates the recurrence properties of mistake dynamical balls in dynamical systems, establishing entropy equivalence, growth rates of return times, and extending results to various systems with applications.

Contribution

It introduces the concept of mistake dynamical balls and proves key recurrence properties, extending previous results to broader classes of dynamical systems.

Findings

Measure-theoretic entropy equals exponential growth rate of return times.

Minimal return times grow linearly with the length of mistake dynamical balls.

Results apply to $eta$-transformations, Axiom A flows, and systems with mild specification.

Abstract

We study the recurrence to mistake dynamical balls, that is, dynamical balls that admit some errors and whose proportion of errors decrease tends to zero with the length of the dynamical ball. We prove, under mild assumptions, that the measure-theoretic entropy coincides with the exponential growth rate of return times to mistake dynamical balls and that minimal return times to mistake dynamical balls grow linearly with respect to its length.Moreover we obtain averaged recurrence formula for subshifts of finite type and suspension semiflows. Applications include $\beta$-transformations, Axiom A flows and suspension semiflows of maps with a mild specification property. In particular we extend some results from [4, 9, 17] for mistake dynamical balls.