Hochman's upcrossing theorem for groups of polynomial growth

Consider a stochastic process $(S_{[a_i,b_i]})_{[a_i,b_i] \subset $\mathbb{N}$}$,whichisindexedbythecollectionofallnonemptyintervals$[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 $k\ge 1$ 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 $(S_{[1,n]}{)}_{n\ge 1}$ 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.