On the asymptotic properties of piecewise contracting maps

Abstract

We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. These maps are Lipschitz continuous, with Lipschitz constant smaller than one, when restricted to any piece of a finite and dense union of disjoint open pieces. We focus on the topological and the dynamical properties of the (global) attractor of the orbits that remain in this union. As a starting point, we show that the attractor consists of a finite set of periodic points when it does not intersect the boundary of a contraction piece, which complements similar results proved for more specific classes of piecewise contracting maps. Then, we explore the case where the attractor intersects these boundaries by providing examples that show the rich phenomenology of these systems. Due to the discontinuities, the asymptotic behaviour is not always properly represented by the dynamics in the attractor. Hence, we introduce generalized orbits to describe the asymptotic dynamics and its recurrence and transitivity properties. Our examples include transitive and recurrent attractors, that are either finite, countable, or a disjoint union of a Cantor set and a countable set. We also show that the attractor of a piecewise contracting map is usually a Lebesgue measure-zero set, and we give conditions ensuring that it is totally disconnected. Finally, we provide an example of piecewise contracting map with positive topological entropy and whose attractor is an interval.

Extremes and Recurrence in Dynamical Systems

“…, N } the restriction f i := f | X i of the map f : X → X of Definition 7.4.4 to a piece X i has the following form: For almost all the values of the parameters a and c, the global attractor Λ is composed of a unique periodic orbit (whose period depends on the specific values of the parameters), but for the remaining set of parameters the global attractor is a Extremal Index -EI set supporting a minimal dynamics. We refer to [248] for a detailed description of the asymptotic dynamics of this map. We just quoted above the case with c = 0 and 0 the (unique) fixed point.…”

Topological dynamics of piecewise -affine maps

“…It is worth observing that the types of maps we consider here appear in the field of diophantine approximation (see [3]). Concerning the dynamics of piecewise contractions, we refer the reader to [5,6,8]. Theorem 1.1 can be reduced to Theorem 1.2, a much more general result.…”

Complexity of injective piecewise contracting interval maps

“…f satisfies P5. Now, Λ is totally disconnected because f satisfies the separation property [5]. It follows that Λ is a Cantor set and f satisfies P3.…”

“…In [5], we explored the diversity of asymptotic dynamics of these systems, and proved that a rich dynamics can appear if the attractor contains discontinuity points. In particular, we exhibited threedimensional examples with exponential complexity and positive topological entropy.…”