Title: Weakly contracting on average RDSs
Abstract.
In this talk, I will present some fundamental properties of random dynamical systems (RDSs) that are weakly contracting on average. This condition will be shown to imply several probabilistic consequences, including the uniqueness of the stationary measure, decay of correlations for Lipschitz observables, concentration inequalities, and an almost sure central limit theorem. I will also illustrate examples to which these results apply, including RDSs on the circle, systems induced by the projective action of linear cocycles, and classically contractive systems of Hutchinson type.
Title: Computer-assisted proof of robust transitivity.
Abstract.
A smooth dynamical system is transitive if it has a dense orbit, loosely meaning that it has some chaos in a topological sense. If this property holds for all diffeomorphisms in a C¹-neighborhood, we say that systems in this neighborhood are robustly transitive. By Bonatti, Diaz and Pujals (2003), robustly transitive diffeomorphisms are volume hyperbolic, and thus they have positive topological entropy, being chaotic in a strict sense and in a robust way. Robust properties are key in classifying smooth dynamical systems, and they are also desirable to model applications.
We develop a computer-assisted strategy to prove robust transitivity in dimension 3 that includes the proof of the existence of a blender. We present a family of D.A. (derived-from-Anosov) systems on the 3-torus for which this strategy applies. This work in preparation is a collaboration with Andy Hammerlindl and Warwick Tucker.
Title: Zip Shift as Extended Symbolic Dynamics: Some of its Applications and Future Perspectives
Abstract.
We will introduce zip shift maps as local homeomorphisms that extend the dynamics of the two-sided shift. Using examples of finite-to-one horseshoe maps, we'll show how these systems can be fully codified, up to topological conjugacy, using symbolic dynamics. We will also explore some applications in ergodic theory, topological dynamics, and cellular automata, and discuss potential future directions for their development and use in these fields.