arXiv subjects

Aimee S. A. Johnson

Publications and source records attributed to Aimee S. A. Johnson.

3 recordsLinked to original sources

arXiv2023

Finite odometer factors of rank one $\mathbb{Z}^d$-actions

In this paper, we give explicit conditions characterizing the Følner rank one $\mathbb{Z}^d$-actions that factor onto a finite odometer; those that factor onto an arbitrary, but specified $\mathbb{Z}^d$-odometer, and those that factor onto an unspecified $\mathbb{Z}^d$-odometer. We also give explicit conditions describing the Følner rank one $\mathbb{Z}^d$-actions that are conjugate to a specific $\mathbb{Z}^d$-odometer, and those that are conjugate to some $\mathbb{Z}^d$-odometer. These conditions are based on cutting and stacking procedures used to generate the action, and generalize results given in \cite{FGHSW} for rank one $\mathbb{Z}$-actions.

math.DS↗

arXiv2021

Topological speedups of $\mathbb{Z}^d$-actions

We study minimal $\mathbb{Z}^d$-Cantor systems and the relationship between their speedups, their collections of invariant Borel measures, their associated unital dimension groups, and their orbit equivalence classes. In the particular case of minimal $\mathbb{Z}^d$-odometers, we show that their bounded speedups must again be odometers but, contrary to the 1-dimensional case, they need not be conjugate, or even isomorphic, to the original.

math.DS↗

arXiv2014

Directional Recurrence for Infinite Measure Preserving Z^d Actions

We define directional recurrence for infinite measure preserving Z^d actions both intrinsically and via the unit suspension flow and prove that the two definitions are equivalent. We study the structure of the set of recurrent directions and show it is always a G_delta set. We construct an example of a recurrent action with no recurrent directions, answering a question posed in a 2007 paper of Daniel J. Rudolph. We also show by example that it is possible for a recurrent action to not be recurrent in an irrational direction even if all its sub-actions are recurrent.

math.DS↗