The pressure function for infinite equilibrium measures

TL;DR

This paper investigates the pressure function for infinite equilibrium measures in dynamical systems, establishing a relation with induced systems and extending prior results to new classes of maps, with applications to statistical properties.

Contribution

It extends Sarig's results to infinite equilibrium states, providing conditions for pressure relations and error estimates, and applies these to various dynamical systems including Pomeau-Manneville maps.

Findings

Pressure relation: $P(+soldsymbol{ ext{ extmu}}) o (C P(ar{+soldsymbol{ extmu}}))^{1/eta}$

Established error bounds for the pressure approximation

Analyzed limit properties and statistical laws of the measures as $s o 0$

Abstract

Assume that $(X,f)$ is a dynamical system and $\phi:X \to [-\infty, \infty)$ is a potential such that the $f$-invariant measure $\mu_\phi$ equivalent to $\phi$-conformal measure is infinite, but that there is an inducing scheme $F = f^\tau$ with a finite measure $\mu_{\bar\phi}$ and polynomial tails $\mu_{\bar\phi}(\tau \geq n) = O(n^{-\beta})$, $\beta \in (0,1)$. We give conditions under which the pressure of $f$ for a perturbed potential $\phi+s\psi$ relates to the pressure of the induced system as $P(\phi+s\psi) = (C P(\overline{\phi+s\psi}))^{1/\beta} (1+o(1))$, together with estimates for the $o(1)$-error term. This extends results from Sarig to the setting of infinite equilibrium states. We give several examples of such systems, thus improving on the results of Lopes for the Pomeau-Manneville map with potential $\phi_t = - t\log f'$, as well as on the results by Bruin & Todd on countably piecewise linear unimodal Fibonacci maps. In addition, limit properties of the family of measures $\mu_{\phi+s\psi}$ as $s\to 0$ are studied and statistical properties (correlation coefficients and arcsine laws) under the limit measure are derived.