Hochman's upcrossing theorem for groups of polynomial growth

TL;DR

This paper extends Hochman's upcrossing theorem, which provides bounds on fluctuations of stationary processes, to groups of polynomial growth, broadening its applicability beyond the integer lattice.

Contribution

The paper generalizes Hochman's upcrossing theorem to groups of polynomial growth, expanding the theorem's scope to a wider class of algebraic structures.

Findings

Established an 'almost-exponential' bound for fluctuations in polynomial growth groups.

Extended techniques from the integer lattice case to more general groups.

Demonstrated applicability of the upcrossing theorem in new algebraic contexts.

Abstract

Consider a stochastic process $(S_{[a_i,b_i]})_{[a_i,b_i] \subset \mathbb{N}}$, which is indexed by the collection of all nonempty intervals $[a_i,b_i] \subset \mathbb{N}$ and which is stationary under translations of the intervals. It was shown by M. Hochman that, for any $k \geq 1$ and any interval $(\alpha,\beta) \subset \mathbb{R}$, one can give an `almost-exponential' bound on the size of the set where the associated process $(S_{[1,n]})_{n \geq 1}$ has at least $k$ fluctuations over $(\alpha,\beta)$. It was also noticed that a similar techniques can be applied in $\mathbb{Z}^d$ case. In this article we extend Hochman's upcrossing theorem to groups of polynomial growth.