We give a discussion of the classical Bowen–Series coding and, in particular, its application to the study of zeta functions and their zeros. In the case of compact surfaces of constant negative curvature κ=−1 the analytic extension of the Selberg zeta function to the entire complex plane is classical, and can be achieved using the Selberg trace formula. However, an alternative dynamical approach is to use the Bowen–Series coding on the boundary at infinity to construct a piecewise analytic expanding map from which the extension of the zeta function can be obtained using properties of the associated transfer operator. This latter method has the advantage that it also applies in the case of infinite area surfaces provided they don’t have cusps. For such examples the location of the zeros is somewhat more mysterious. However, in particularly simple cases there is a striking structure to the zeros when we take appropriate rescaling. We give some insight into this phenomenon.
This is a mini-course taught by the author at IMPAN in May 2023. It discussed the following circle of questions: Let X be an open simply connected Riemann surface with some concrete geometric description, and ϕ a uniformizing function. How are the geometric properties of X related to the properties of ϕ? Classical and modern results are discussed, and some proofs are included.
This paper is partly an exposition, and partly an extension of our work [Adv. Math. 399 (2022)] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of ergodic measures for parametrized families of smooth iterated function systems (IFS) on the line. Under some assumptions, most crucially, a transversality condition, we obtain formulas for the Hausdorff dimension of the measure and absolute continuity for almost every parameter in the appropriate parameter region. The main novelty of loc. cit. and the present paper is that not only the IFS, but also the ergodic measure in the symbolic space, whose push-forward we consider, depends on the parameter. This includes many interesting families of measures, in particular, invariant measures for IFS’s with place-dependent probabilities and natural (equilibrium) measures for smooth IFS’s. One of the goals of this paper is to present an exposition of loc. cit. in a more reader-friendly way, emphasizing the ideas and proof strategies, but omitting the more technical parts. This exposition/survey is based in part on the series of lectures by Károly Simon at the Summer School “Dynamics and Fractals” in 2023 at the Banach Center, Warsaw. The main new feature, compared to loc. cit., is that we consider multi-parameter families; in other words, the set of parameters is allowed to be multi-dimensional. This broadens the scope of applications. A new application considered here is to a class of Furstenberg-like measures.
These notes correspond to two separate mini-lecture courses presented at the Banach Centre, Warsaw, in the Spring of 2023. However, they are united by a common theme which was the study of hyperbolic dynamical systems and ideas from thermodynamic formalism and its applications.
Lyapunov exponents are fundamental invariants in smooth ergodic theory describing the asymptotic infinitesimal behavior along typical orbits. This text aims to explain how and why to control Lyapunov exponents using entropy for smooth surface diffeomorphisms. It fits into the framework of our recent joint works with Sylvain CROVISIER and Omri SARIG. We will focus especially on the continuity property of exponents for measures near the maximal entropy measure, by presenting a simplified version of the original argument. Our exposition is geared towards advanced students and researchers in dynamics that are not necessarily familiar with smooth ergodic theory.
