On the approximation of finite and infinite equilibrium states and some aspects of young towers with non-integrable return function

It is well known that for different classes of transformations, including the class ofpiecewise C2 expanding maps T : [0, 1] ª, Ulam’s method is an efficient way tonumerically approximate the absolutely continuous invariant measure of T. We developa new extension of Ulam’s method and prove that this extension can be usedfor the numerical approximation of the Ruelle-Perron-Frobenius operator associatedwith T and the potential Á¯ = −¯ log |Tp|, where ¯ 2 R. In particular we provethat our extended Ulam’s method is a powerful tool for computing the topologicalpressure P(T, Á¯) and the density of the equilibrium state. This material is work incollaboration with Gary Froyland and main part of it has appeared in [62])Taking one step further, we demonstrate numerically that this extension is a veryefficient tool for the numerical approximation of thermodynamical objects such astopological pressure, conformal measures and equilibrium states associated with nonuniformlyexpanding systems such as the Farey map ˜ T. This material is work incollaboration with Gary Froyland and Rua Murray and has appeared in [20]).To complete the picture of phase transition in the Farey map case, we show thatthe results of Sarig [59, 60] offer a rigorous proof of the informal statements madeby Prellberg and Slawny [53] on the bifurcation of the equilibrium states associatedwith ( ˜ T,−¯ log | ˜ Tp); this also allows one to state rigorously that the infinite, ¾-finite˜ T-Lebesgue a.c.i.m. (corresponding to ¯ = 1) can be interpreted as a weak ES in thesense of [60].One method used to study statistical properties of (probability measure preserving)non-uniformly hyperbolic systems is to derive them from the statistical propertiesof their corresponding Young towers (YTs), when they exists. Our main aim in thispart of the thesis is to formulate necessary and sufficient conditions that allows one toinfer distributional convergence results for infinite measure preserving systems fromproperties of their corresponding Young towers. As we will explain, these conditionsare exactly the ones that relate the infinite, ¾-finite invariant measure of the transformationin question to the ¾-finite invariant measure of the corresponding tower.In this part, we will also point out some interesting consequences of these conditionsfor the asymptotic tail behavior of the YT associated with an infinite m.p.t. that donot follow in the probability case. Material related to this part of the thesis is workin progress with Henk Bruin and Matthew Nicol(see [8])