Escape rates for Gibbs measures (Q2908167)

The authors study the asymptotic behavior of the escape rate of Gibbs measures, which are supported in a conformal repeller. This problem is more specifically posed in the following way: let \(f : X \to X\) be a map preserving an ergodic probability measure \(\mu\). The escape rate of \(\mu\) through the hole \(U \subset X\) is the rate at which mass is lost through \(U\), quantifying the asymptotic behavior of the \(\mu\)-measure of the set of the points \(x \in X\) for which none of the first orbits of \(x\) intersects \(U\). Thus the issue is to evaluate the average NEWLINE\[NEWLINER_\mu(U)=\limsup_{k\to \infty} \frac1k \log\mu (\{ x : f^i (x) \notin U,\;i = 0, 1,\dots, k -1\})NEWLINE\]NEWLINE The setting in which the problem is treated is the following: Let \(f:M\to M\) be a \(C^1\) map, with \(M\) a Riemannian manifold. Let \(J\subset M\) be a conformal repeller such that \(f\) is topologically mixing in \(J\). If \(\varphi:J\to \mathbb R\) is a Hölder continuous map and \(z\in J\) then the main result obtained by the authors is NEWLINE\[NEWLINE\lim_{\varepsilon\to0}\frac{R_\mu(B_\varepsilon(z))}{\mu(B_\varepsilon(z))}=\begin{cases}1&\text{if \(z\) is not periodic}\\ 1-\exp(S_p(z)-pP(\varphi))&\text{if \(z\) has a prime period \(p\)},\end{cases}NEWLINE\]NEWLINE where \(S_p(z)=\sum_{i=0}^{p-1}\varphi(f^i(z))\) and \(P(\varphi)\) is the topological pressure at \(\varphi\).NEWLINENEWLINEThe problem is considered in the level of subshifts \((\Sigma,\sigma)\). One of the key arguments to obtain this result is the analysis of the spectral properties of the Ruelle operator \(\mathcal L\) and the perturbed operators \(\mathcal L_n\). These last operators are defined from a certain family of sets \((U_n)\), and have eigenvalues \(\lambda_n>0\) and eigenmeasures \(\nu_n\) supported in the set \(V_n :=\bigcap_{k\geq 0}(\sum\sigma^{-k}(U_n))\). Another important result used by the authors is the existence of Gibbs measures for conformal repellers.NEWLINENEWLINEIt is pointed out that NEWLINE\[NEWLINER_\mu(U_n)=-\log \lambda_n,NEWLINE\]NEWLINE and from this and several important spectral properties of the perturbed operators established in this paper, the desired result is obtained.NEWLINENEWLINEAlso, the authors show an asymptotic formula for the Hausdorff dimension of the set \(\mu (\{x:f^i(x)\notin (B_\varepsilon(z))\text{ for any }i \geq 0\})\).