Entropy for Generalized Parabolic Dynamics

TL;DR

This paper extends the concept of generalized entropy to uniform spaces and applies it to parabolic dynamics, introducing a broader class called generalized parabolic dynamics and analyzing their entropy properties.

Contribution

It introduces generalized entropy for uniform spaces, characterizes a subclass of generalized parabolic dynamics, and constructs examples with prescribed entropy growth rates.

Findings

Identified a subclass of generalized parabolic dynamics with linear entropy.

Proved equivalences linking generalized entropy, non-wandering sets, and singular subsets.

Constructed homeomorphisms with prescribed entropy growth and singleton non-wandering set.

Abstract

In this paper, we extend the concept of generalized entropy to uniform spaces, allowing computations beyond metrizable settings. We apply this to parabolic dynamics - systems with a unique fixed point uniformly attracting all compact subsets in both time directions - and introduce a broader class, called generalized parabolic dynamics. Within this, we identify a significant subclass and prove its linear entropy, offering several equivalent characterizations linking the generalized entropy of the space, the non-wandering set, and families of mutually singular subsets. We also study homeomorphisms of compact surfaces with a singleton non-wandering set but non-parabolic dynamics. The examples that we present have at least quadratic entropy, bounded above by the supremum of polynomial growth rates. For any growth rate with the linear invariant property within these bounds, we construct a homeomorphism whose generalized entropy realizes the prescribed growth, with the non-wandering set reduced to a point.