Topological Dynamics of Highly Irregular Maps

The present dissertation is devoted to the study of discontinuous dynamical systems from the perspective of topological dynamics. Motivated by the prevalence of discontinuities in natural and symbolic processes and by the interest in topological dynamics to understand dynamical behavior beyond the classical framework of continuity, we investigated a variety of settings in which discontinuities are “abundant” from a metric-topological perspective. A common setting is a densely distributed set of discontinuity points in the space. The thesis is organized into four main directions, linked by the common aim of exploring which known dynamical results for continuous maps remain recoverable, possibly in a suitable, weaker form with less regularity. First, we analyze open dynamical systems, in which orbits may escape through shrinking families of holes. Open systems can be seen as simple cases of discontinuous systems because the “holes” behave in a different way with respect to any vicinity. We show that indecisive behavior (orbits switching infinitely often between escape routes) is a generic phenomenon for typical transitive homeomorphisms, revealing a novel form of dynamical instability. Second, we study Newton’s method as a discrete dynamical system on real intervals. More precisely, given a function f: [a, b] → R , for any point x in the real interval , we consider the trajectory 0 [a, b] {x n } n=0 ∞ generated by Newton's method, that is, where x . We prove n+1 = N(f, x n ) = x n − f(x n )/f'(x n ) that for the typical continuously differentiable function f, a chaotic behavior, in the sense of Devaney, of N(f, ·) is necessarily confined to nowhere dense sets, whereas under weaker smoothness assumptions of f, typically the trajectories exhibit surprisingly convergence (with uniform rate) to the roots of f, highlighting the subtle interplay between regularity and global dynamics. Third, we focus on classes of interval maps defined by symbolic and combinatorial constructions, including erasing substitutions and critical exponent functions. These systems, characterized by dense discontinuities, exhibit rich chaotic properties. Finally, the thesis addresses the structure of chain-recurrence and chain components in compact dynamical systems with and without continuity assumptions. We show, among other results, that even in the absence of regularity, every compact system admits a chain-recurrent point and a closed invariant chain-transitive subsystem. Together, these investigations highlight the dynamical richness that arises when continuity is abandoned, contributing to a broader understanding of topological dynamics in minimal regularity contexts.