Weak mixing behavior for the projectivized derivative extension

TL;DR

This paper constructs smooth and analytic diffeomorphisms with zero topological entropy exhibiting weak mixing in the projectivized tangent bundle, expanding understanding of ergodic properties in low-entropy systems.

Contribution

It introduces a new construction method for weakly mixing diffeomorphisms with zero entropy on manifolds with circle actions, including the 2-torus.

Findings

Existence of weakly mixing diffeomorphisms with zero entropy

Construction of analytic examples on the 2-torus

Application of a quantitative approximation by conjugation method

Abstract

In both smooth and analytic categories, we construct examples of diffeomorphisms of topological entropy zero with intricate ergodic properties. On any smooth compact connected manifold of dimension 2 admitting a nontrivial circle action, we construct a smooth diffeomorphism whose differential is weakly mixing with respect to a smooth measure in the projectivization of the tangent bundle. In case of the 2-torus, we also obtain the analytic counterpart of such a diffeomorphism. The constructions are based on a quantitative version of the ``Approximation by Conjugation'' method, which involves explicitly defined conjugation maps, partial partitions, and the adaptation of a specific analytic approximation technique.