Keplerian shear for Chacon Transformations

TL;DR

This paper develops a new approach to keplerian shear for systems with weakly mixing fibers, applying it to a family of random Chacon-like transformations and proving they exhibit keplerian shear using a local limit theorem.

Contribution

It introduces an approach for systems with weakly mixing fibers and demonstrates keplerian shear in a new class of random rank one transformations.

Findings

Proves keplerian shear for a family of Chacon-type transformations.

Develops a local limit theorem for time-dependent Birkhoff sums.

Extends the concept of keplerian shear to systems with weakly mixing fibers.

Abstract

The concept of keplerian shear was introduced by Damien Thomine recently. It is useful for non ergodic systems, and can be seen as strong mixing conditionally on invariant fibers. The notion is particularly interesting when a.e. fiber is not strongly mixing. We develop here an approach appropriate for systems such that a.e. fiber is weakly mixing, and apply it to a family of rank one transformations. Each transformation is a kind of Chacon map, built with a random number of spacers at each step of the Rochklin tower. We prove that this new dynamical system exhibits keplerian shear. The method relies on a version of a local limit theorem for time dependent Birkhoff sums along the fullshift.