Complexity of injective piecewise contracting interval maps

Abstract

We study the complexity of the itineraries of injective piecewise contracting maps on the interval. We prove that for any such map the complexity function of any itinerary is eventually affine. We also prove that the growth rate of the complexity is bounded from above by the number N − 1 of discontinuities of the map. To show that this bound is optimal, we construct piecewise affine contracting maps whose itineraries all have the complexity (N − 1)n + 1. In these examples, the asymptotic dynamics takes place in a minimal Cantor set containing all the discontinuities.

Symbolic dynamics of piecewise contractions

“…1 and 2.2 imply, in particular, the result of Catsigeras, Guiraud and Meyroneinc[7] concerning the complexity function of languages of n-PCs, which is stated below in a more complete way, with f T given by Theorem 2.2.…”

“…Natural codings of piecewise contractions defined on 2 intervals (or more generally, defined on 2 complete metric spaces) were provided by Gambaudo and Tresser [14] and are intrinsically related to natural codings of rotations of the circle. Concerning languages of injective n-PCs f : I → I for n > 2, some progress was made recently by Catsigeras, Guiraud and Meyroneinc [7]. They proved that for each natural coding θ of f , the complexity function of the language L(θ), defined by p θ (k) = #L k (θ), where # denotes cardinality, is eventually affine.…”

A spectral decomposition of the attractor of piecewise-contracting maps of the interval

“…It is in particular the case for the half-closed unit interval map x → λx + µ mod 1, for which the properties of the rotation number as a function of λ and µ ∈ [0, 1) have been studied in detail [2,3,6,8]. For injective PCIMs with N 2 contraction pieces, it has been proved that the complexity of the itinerary of any orbit is an eventually affine function [4,13]. The growth rate of the complexity is at most equal to N − 1 and there are some examples of PCIMs with such a maximal complexity [4].…”

“…For injective PCIMs with N 2 contraction pieces, it has been proved that the complexity of the itinerary of any orbit is an eventually affine function [4,13]. The growth rate of the complexity is at most equal to N − 1 and there are some examples of PCIMs with such a maximal complexity [4]. In these particular examples, the attractor is a minimal Cantor set containing all the boundaries of the contraction pieces.…”

“…In [4], it is shown that for injective PCIMs the complexity of the itinerary of any point in X is an eventually constant or affine function. As a consequence of Theorem 1.1, we obtain that if D ⊂ X then the ω-limit sets of the points with affine complexity are Xminimal Cantor sets.…”