Bohr chaoticity, semi-horseshoes and full-entropy abundance

TL;DR

This paper establishes that systems with semi-horseshoes exhibit Bohr chaoticity, linking topological entropy, invariant measures, and complexity, with broad implications across dynamical systems.

Contribution

It proves that systems with semi-horseshoes are Bohr chaotic and explores entropy properties of correlated point sets in various dynamical contexts.

Findings

Systems with semi-horseshoes are Bohr chaotic.

Correlated point sets have positive or full topological entropy.

Results apply to algebraic, smooth, and generic topological systems.

Abstract

Bohr chaoticity is a topological notion of dynamical complexity defined through non-orthogonality to all non-trivial weights. It is strictly stronger than positivity of topological entropy and also has strong consequences for the invariant-measure structure. In this paper, we show that every dynamical system having a semi-horseshoe, including every positive-entropy graph map and every $C^1$ partially hyperbolic diffeomorphism, is Bohr chaotic; furthermore, the set of points correlated with any given non-trivial weight has positive topological entropy. Moreover, for positive-entropy dynamical systems with either the shadowing property or the modified almost specification property, such set can has full topological entropy. Our results also yield applications in several classical algebraic and smooth settings, as well as in the $C^0$-generic setting of topological dynamics.