On the Complexity of some Geometrical Objects

TL;DR

This paper investigates the $\epsilon$-distortion complexity of Cantor sets and measures, providing new bounds and extending the concept to dynamical systems, enhancing understanding of their geometric and measure-theoretic complexity.

Contribution

It introduces an analogous $\epsilon$-distortion complexity for measures and establishes new lower bounds for analytic iterated function systems, along with upper bounds for hyperbolic dynamical systems.

Findings

New lower bounds for Cantor sets defined by analytic IFS

Upper bounds for invariant sets in hyperbolic systems

Extension of complexity concepts to measures

Abstract

We recall the definition of the $\epsilon$-distortion complexity of a set defined in \cite{bcc} and the results obtained in this paper for Cantor sets of the interval defined by iterated function systems. We state an analogous definition for measures which may be more useful when dealing with dynamical systems. We prove a new lower bound in the case of Cantor sets of the interval defined by analytic iterated function systems. We also give an upper bound the $\epsilon$-distortion complexity of invariant sets of uniformly hyperbolic dynamical systems.