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

Abstract

Let $(X_k{)}_{k\ge 0}$ 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 $\frac{1}{n}\log R_n$ 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 \[ \mathcal{R}(q)=\lim_n\frac{1}{n}\log\int R_n^q \,dμ\quad (q\in\mathbb{R}) \] provided the limit exists. We consider the case where $(X_k{)}_{k\ge 0}$ is distributed according to the equilibrium state of a potential $\phi :{\mathcal{A}}^{\mathbb{N}}\to \mathbb{R}$ with summable variation, and we prove that \[ \mathcal{R}(q) = \begin{cases} P((1-q)φ) & \text{for}\;\; q\geq q_φ^*\\ \sup_η\int φ\, dη& \text{for}\;\; q<q_φ^{*} \end{cases} \] where $P((1-q)\phi )$ is the topological pressure of $(1-q)\phi$, the supremum is taken over all shift-invariant measures, and $q_{\phi }^{\ast }$ is the unique solution of $P((1-q)\phi )={\sup }_{\eta }\int \phi d\eta$. Unexpectedly, this spectrum does not coincide with the $L^q$-spectrum of ${\mu }_{\phi }$, which is $P((1-q)\phi )$, and does not coincide with the waiting-time $L^q$-spectrum in general. In fact, the return-time $L^q$-spectrum coincides with the waiting-time $L^q$-spectrum if and only if the equilibrium state of $\phi$ is the measure of maximal entropy. As a by-product, we also improve the large deviation asymptotics of $\frac{1}{n}\log R_n$.

29 pages, 1 figure, submitted. This is a completely new version. All statements in the previous version are correct, but the proof of the main result relied on a result which turned out to be false. This is now fixed and gave rise to a companion paper by three of the present authors, see arXiv:2101.12381

References in corpus (4)

• Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions
• Sharp error terms for return time statistics under mixing conditions
• Generalized dimensions, large deviations and the distribution of rare events
• On Large Deviation Property of Recurrence Times

Cited by in corpus (1)

• Number of visits in arbitrary sets for $\varphi$-mixing dynamics