Construction of some new diffeomorphisms using a combinatorial approach: the approximation by conjugation method

Abstract

Ergodic theory often involves dealing with transformations of measure spaces that preserve specific measures, such as volume or probability measures. These transformations establish a framework for studying the behavior of dynamical systems. Combinatorial methods are pivotal in ergodic theory, providing valuable techniques for understanding the behavior of dynamical systems. They are often utilized in the study of measure-preserving transformations, aiding in characterizing invariant measures, exploring ergodic decomposition, and establishing connections between different dynamical systems. Moreover, numerous scenarios exist where we can effectively characterize a measure-preserving transformation and its long-term behaviour by approaching it as the limit of specific finite objects, such as periodic processes. This thesis discusses combinatorial constructions within Ergodic Theory, employing the “Approximation by Conjugation” method. This technique facilitates the construction of maps with specific topological and measure-theoretic characteristics on the manifolds that support a non-trivial circle action. We provide examples of volume-preserving diffeomorphisms with zero topological entropy. These examples exhibit intricate ergodic properties in both smooth and analytic categories. Specifically, we present an example of a smooth diffeomorphism with an invariant measure, which is a generic but non-ergodic volume measure. This diffeomorphism satisfies various other ergodic and topological dynamic properties on the 2-torus. In ergodic theory, generic points are essential for understanding the statistical and dynamical behavior of systems, as their orbits cover most of the phase space according to the invariant measure. We examine the distinctions between generic and non-generic points in smooth dynamical systems, focusing on establishing bounds for the sizes of their respective sets. We construct an explicit collection of sets that encompass all generic points within the system and focus on determining bounds for their Hausdorff dimension, leading to more insightful and compelling conclusions. Additionally, we explore the ergodic properties of diffeomorphisms and examine their differential maps with respect to a smooth measure in the projectivization of the tangent bundle. We investigate examples of diffeomorphisms with complex ergodic properties within the projectivized tangent bundle for the smooth and analytic case. We construct a smooth diffeomorphism whose differential is weakly mixing with respect to a smooth measure in the projectivization of the tangent bundle. In the case of the 2-torus, we also obtain the analytic counterpart of such a diffeomorphism by utilizing an analytic approximation technique. We present an example of an analytic diffeomorphism whose projectivized derivative extension exhibits weak mixing with respect to a smooth measure. All of these constructions are based on the original and quantitative version of the “Approximation by Conjugation” method. This approach utilizes highly explicit well-defined conjugation maps, explicit partial partitions of the space, specific approximation strategies, and geometric and combinatorial criteria to meet specific requirements.