On the Dynamics of Induced Maps on the Space of Probability Measures

TL;DR

This paper investigates the complex dynamics of induced maps on probability measures for generic continuous maps and homeomorphisms on the Cantor space, focusing on chaos, entropy, and recurrence properties.

Contribution

It provides new insights into the behavior of induced maps on probability measures for generic maps and extends some results to arbitrary compact metric spaces.

Findings

Analysis of Li-Yorke chaos in induced maps

Results on topological entropy and recurrence

Characterization of equicontinuity and shadowing properties

Abstract

For the generic continuous map and for the generic homeomorphism of the Cantor space, we study the dynamics of the induced map on the space of probability measures, with emphasis on the notions of Li-Yorke chaos, topological entropy, equicontinuity, chain continuity, chain mixing, shadowing and recurrence. We also establish some results concerning induced maps that hold on arbitrary compact metric spaces.