We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the $ \zeta $-Hölder regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $ \delta $, we show that the $ F $-invariant measure varies continuously with respect to a suitable anisotropic norm. Furthermore, we establish that certain interesting classes of perturbations exhibit a modulus of continuity estimated by $ D_2\delta^\zeta \log \delta $, where $ D_2 $ is a constant.

The notion of expansivity and its generalizations (measure expansive, measure positively expansive, continuum-wise expansive, countably-expansive) are well known for deterministic systems and can be a useful property for studying significant type of behavior, such as chaotic one. This study aims to extend these notions into a random context and prove a relationship between relative positive entropy and random expansive measures and apply it to show that if a random dynamical system has positive relative topological entropy then the w-stable classes have zero measure for the conditional measures. We also prove that there exists a probability measure that is both invariant and expansive. Moreover, we obtain a relation between the notions of random expansive measures and random countably-expansive systems.

We consider a robust class of random non-uniformly expanding local homeomorphisms and Hölder continuous potentials with small variation. For each element of this class we develop the thermodynamical formalism and prove the existence and uniqueness of equilibrium states among non-uniformly expanding measures. Moreover, we show that these equilibrium states and the random topological pressure vary continuously in this setting.

We investigate the spectral types of families of discrete one-dimensional Schrödinger operators { H ω } ω ∈ Ω , where each H _ω has a potential function V _ω (n) = f(T ⁿ ω) for n ∈ Z , T is an ergodic homeomorphism on a compact space Ω and f : Ω → R is a continuous function. The main analytical tool of the paper is Gordon’s lemma, of which we provide a complete proof. We demonstrate that a generic operator H ω ∈ { H ω } ω ∈ Ω has purely continuous spectrum if { T n α } n ≥ 0 is dense in Ω for a certain α ∈ Ω. Furthermore, we extend this result to the case where {Ω, T} satisfies topological repetition property, a concept introduced by Boshernitzan and Damanik (2008, [1]). Theorems presented in this paper weaken the hypotheses of previous research and enable us to reach the same conclusion as those authors. Our paper presents a comprehensive study of the topological and metric repetition properties, highlighting their binding role between the theory of dynamical systems and spectral theory. We also discuss the implications of these properties on the qualitative description of specific physical systems.

Estas notas de clase son el resultado de la experiencia de los autores, tanto estudiantes como profesores, en el curso de Teoría de la medida o Medida e integración. Son notas básicas para estudiantes de programas de pregrado en matemáticas y áreas afines a las matemáticas. La parte teórica es una recopilación de la teoría estudiada en los libros clásicos. Con la publicación de estos apuntes, nuestra contribución es presentar un libro de fácil comprensión. Por ello, se han añadido muchos diagramas y hemos escrito los detalles de las pruebas y ejemplos que aparecen a lo largo de la teoría. Al final de cada capítulo hay una lista de ejercicios con la idea principal de su solución para practicar la teoría que se acaba de estudiar.