Equilibrium States for Open Zooming Systems
Eduardo Santana
TL;DR
We address open zooming systems, where a hole H renders part of the phase space noninvariant, and develop an inducing-scheme approach to code the dynamics via a finite Markov structure with infinitely many symbols. For zooming potentials with finite pressure and locally Hölder induced potentials, we prove the existence of finitely many ergodic equilibrium states that are zooming measures; for pseudo-geometric potentials $\phi_t = -t \log J_{\mu} f$, we obtain pseudo-conformal measures and, under mild conditions, uniqueness of the equilibrium state without requiring transitivity. The core method lifts the dynamics to a countable Markov shift, applies Sarig's thermodynamic formalism, and projects equilibrium states back to the original system, with Abramov-type relations connecting lifted and base systems. The results apply to important examples such as Viana maps and extend equilibrium-state theory to open, nonuniformly expanding settings with nonexponential contraction, providing a robust framework for understanding mass escape and statistical behavior in open dynamical systems.
Abstract
In this work, we construct Markov structures for zooming systems adapted to holes of a special type. Our construction is based on backward contractions provided by zooming times. These Markov structures may be used to code the open zooming systems. In the context of open zooming systems, possibly with the presence of a critical/singular set, we prove the existence of finitely many ergodic zooming equilibrium states for zooming potentials whose induced potential is locally Hölder. For example, the zooming Hölder continuous. Among the zooming ones are the so-called \textit{hyperbolic potentials} and also what we call \textit{bounded distortion potentials}, having as a particular case the \textit{pseudo-geometric potentials} $φ_{t} = -t \log J_μf $, where $J_μf$ is a Jacobian of the reference zooming measure. Moreover, for this last class of potentials, we show the existence of what we call pseudo-conformal measures. Afterwards, with a mild condition, we prove uniqueness of equilibrium state with no requirement of transitivity. The technique consists in using an inducing scheme in a finite Markov structure with infinitely many symbols to code the dynamics to obtain an equilibrium state for the associated symbolic dynamics and then projecting it to obtain an equilibrium state for the original map. To obtain a pseudo-conformal measure, we "spread" the conformal one which exists for the symbolic dynamics. The uniqueness is obtained by showing that the equilibrium states are liftable to the same inducing scheme. Finally, we show that the class of hyperbolic potentials is equivalent to the class of continuous zooming potentials. Moreover, we give a class of examples of hyperbolic potentials (including the null one). It implies the existence and uniqueness of equilibrium state. Among the maps considered is the important class known as Viana maps.