Generic mixing is rank-1

TL;DR

This paper proves that in the space of mixing transformations, the conjugacy class of any mixing transformation is dense, leading to the conclusion that generic mixing transformations are rank-1.

Contribution

It establishes that for any mixing transformation, its conjugacy class is dense in the space of mixing transformations with leash-topology, confirming that generic mixing is rank-1.

Findings

Conjugacy classes of all mixing transformations are dense.

Generic mixing transformations are rank-1.

The result applies to the space of mixing transformations with leash-topology.

Abstract

In paper [1] S. V. Tikhonov introduced a complete metric on the space of mixing transformations. It generates a topology called leash-topology. In [2] Tikhonov states the following problem: for what mixing transformation T its conjugacy class is dense in the space of mixing transformations with leash-topology? We show that it is true for every mixing T. As a corollary we get that generic mixing is rank-1.