Piecewise contracting maps on the interval: Hausdorff dimension, entropy and attractors

TL;DR

This paper investigates the attractor of a piecewise contracting map on an interval, demonstrating that both the topological entropy and Hausdorff dimension are zero when the map is injective, linking these properties to orbit complexity.

Contribution

It provides a method to estimate entropy and Hausdorff dimension for injective piecewise contracting maps, showing both are zero, which advances understanding of their dynamical complexity.

Findings

Topological entropy of the system is zero.

Hausdorff dimension of the attractor is zero.

Orbit complexity determines these properties.

Abstract

We consider the attractor $\Lambda$ of a piecewise contracting map $f$ defined on a compact interval. If $f$ is injective, we show that it is possible to estimate the topological entropy of $f$ (according to Bowen's formula) and the Hausdorff dimension of $\Lambda$ via the complexity associated with the orbits of the system. Specifically, we prove that both numbers are zero.