Bohr chaoticity, semi‐horseshoes, and full‐entropy abundance

Abstract

Bohr chaoticity is a topological notion of dynamical complexity defined through nonorthogonality to all nontrivial 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 partially hyperbolic diffeomorphism, is Bohr chaotic; furthermore, the set of points correlated with any given nontrivial weight has positive topological entropy. Moreover, for positive‐entropy dynamical systems with either the shadowing property or the modified almost specification property, the set has full topological entropy. Our results also yield applications in several classical algebraic and smooth settings, as well as in the ‐generic setting of topological dynamics.