[Paper Review] Piecewise contractions defined by iterated function systems

Arnaldo Nogueira, Benito Pires+1 more|arXiv (Cornell University)|Aug 7, 2014

Mathematical Dynamics and Fractals10 references3 citations

TL;DR

This paper establishes that for Lebesgue almost every choice of partition points in [0,1], an n-interval piecewise contraction defined by an iterated function system (IFS) of Lipschitz contractions is asymptotically periodic, meaning every orbit converges to a periodic orbit. The key result is that such maps have between one and n periodic orbits, and the dynamics are fully characterized in terms of the IFS's contraction properties and the structure of nested sets A_k.

ABSTRACT

Let $ϕ_1,\ldots,ϕ_n:[0,1] o (0,1)$ be Lipschitz contractions. Let $I=[0,1)$, $x_0=0$ and $x_n=1$. We prove that for Lebesgue almost every $(x_1,...,x_{n-1})$ satisfying $0

Motivation & Objective

• To characterize the topological dynamics of piecewise contractions (PCs) defined by iterated function systems (IFSs) in full generality, including non-injective maps.
• To extend prior results—previously restricted to injective IFSs with non-overlapping ranges—by developing a framework applicable to both injective and non-injective contractions.
• To prove that for almost every partition of [0,1], the resulting PC map is asymptotically periodic, i.e., every orbit converges to a periodic orbit.
• To establish tight bounds on the number of periodic orbits, showing that it is at most n for such typical maps.

Proposed method

• Define the map f as a piecewise contraction using n Lipschitz contractions φ_i and a partition of [0,1) into n intervals defined by points x_1 < ... < x_{n-1}.
• Introduce the nested sequence of sets A_k, where A_0 = [0,1] and A_k = ∪_{i=1}^n φ_i(A_{k-1}) for k ≥ 1, to analyze the asymptotic behavior.
• Prove that if the IFS is highly contracting (or κ-Lipschitz with κ < 1/2), then ∩_{k≥0} A_k is a null set, which implies asymptotic periodicity.
• Use combinatorial arguments on a quasi-partition of intervals to bound the number of periodic orbits by n, based on equivalence classes of intervals under a defined relation.
• Apply measure-theoretic arguments to show that the set of parameter configurations (x_1,…,x_{n-1}) for which the dynamics are not asymptotically periodic has Lebesgue measure zero.
• Construct a counterexample (affine maps φ_1, φ_2 with κ > 1/2) to show that ∩A_k can have positive measure, demonstrating the necessity of the contraction condition.

Experimental results

Research questions

• RQ1 Under what conditions on the IFS {φ_1,…,φ_n} is the associated piecewise contraction asymptotically periodic for almost every partition of [0,1)?
• RQ2 What is the maximum number of periodic orbits that a typical n-interval piecewise contraction can have, and is this bound tight?
• RQ3 How does the structure of the nested sets A_k = ∪φ_i(A_{k-1}) relate to the long-term dynamics of the map f?
• RQ4 Can the results for injective IFSs be extended to non-injective maps with overlapping ranges, and if so, under what conditions?
• RQ5 What happens to the dynamics when the IFS is not highly contracting—can ∩A_k still be a null set, or can it have positive measure?

Key findings

• For Lebesgue almost every (x_1,…,x_{n-1}) ∈ Ω_{n-1}, the piecewise contraction f defined by the IFS is asymptotically periodic, meaning ω_f(x) is a periodic orbit for every x ∈ [0,1).
• The number of periodic orbits of such a typical f is at least one and at most n, with the upper bound being sharp.
• The asymptotic periodicity is guaranteed if the IFS consists of κ-Lipschitz contractions with κ < 1/2, a weaker condition than being highly contracting.
• The set ∩_{k≥0} A_k is a null set for typical IFSs under the κ < 1/2 condition, which ensures that the dynamics eventually enter a periodic regime.
• A counterexample is constructed where ∩A_k = [1/8, 1/2], a non-degenerate interval, showing that the null set condition fails when κ ≥ 1/2.
• The number of periodic orbits is bounded by the number of equivalence classes in a combinatorially defined relation on intervals, which is at most n, proving the upper bound.