The complete $L^q$-spectrum and large deviations for return times for equilibrium states with summable potentials

M. Abadi, J. -R. Chazottes, S. Gallo

Published 2019-02-09Version 1

Let $\{X_k\}_{k\geq 1}$ be a stationary and ergodic process with joint distribution $\mu$ where the random variables $X_k$ take values in a finite set $\mathcal{A}$. Let $R_n$ be the first time this process repeats its first $n$ symbols of output. It is well-known that $n^{-1} \log R_n$ converges almost surely to the entropy of the process. Refined properties of $R_n$ (large deviations, multifractality, etc) are encoded in the return-time $L^q$-spectrum defined as \[ \mathcal{R}(q)=\lim_n\frac{1}{n}\log\int R_n^q\, d\mu \] provided the limit exists. When $\{X_k\}_{k\geq 1}$ is distributed according to the equilibrium state of a potential $\varphi$ with summable variation, we are able to compute $\mathcal{R}(q)$ for all $q\in\mathbb{R}$, namely \[ \mathcal{R}(q) = \begin{cases} P((1-q)\varphi) & \text{for}\;\; q\geq q^*\\ \sup_\eta \int \varphi \, d\eta & \text{for}\;\; q\le q^* \end{cases} \] where the supremum is taken over all shift-invariant probability measures, $P((1-q)\varphi)$ is the topological pressure of $(1-q)\varphi$ ($q\in\mathbb{R})$, and $q^*=q^*_\varphi\in [-1,0)$ is the unique solution of $P((1-q)\varphi) =\sup_\eta \int \varphi \, d\eta$, as $q$ runs through $\mathbb{R}$. Surprisingly, this spectrum does not coincide with the hitting-time $L^q$-spectrum for all $q<q^*$. The two spectra coincide if and only if the equilibrium state of $\varphi$ is the measure of maximal entropy. As a by-product, we prove large deviation asymptotics for $n^{-1}\log R_n$.