Symbolic dynamics of piecewise contractions

Author(s)

Pires, Benito

• Document type: Journal article
• Source: Nonlinearity; v. 32, n. 12, p. 4871-4889, DEC 2019.
• Web of Science Citations: 0

Abstract

A map f : {[}0, 1) -> {[}0, 1) is a piecewise contraction of n intervals (n-PC) if there exist 0 < lambda < 1 and a partition of I = {[}0, 1) into intervals I-1, I-2,..., I-n such that vertical bar f (x) - f (y)vertical bar <= lambda vertical bar x - y vertical bar for every x, y is an element of I-i (i = 1, 2,..., n). An infinite word theta = theta(0)theta(1)... over the alphabet A = [1,..., n] is a natural coding of f if there exists x is an element of I such that theta k = i whenever f(k)(x) is an element of I-i. We prove that if theta is a natural coding of an injective n-PC, then some infinite subword of theta is either periodic or isomorphic to a natural coding of a topologically transitive m-interval exchange transformation (m-IET), where m <= n. Conversely, every natural coding of a topologically transitive n-IET is also a natural coding of some injective n-PC. (AU)

• FAPESP's process: 18/06916-0 - Chaotic dynamical systems
• Grantee: Benito Frazao Pires
• Support Opportunities: Regular Research Grants