All solenoids of piecewise smooth maps are period doubling

Lluís Alsedà; Víctor Jiménez López; L’ubomír Snoha

ArticleOriginal scientific text

Title

All solenoids of piecewise smooth maps are period doubling

Authors ¹, ², ³

Affiliations

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, Aptdo. de Correos 4021, 30100 Murcia, Spain
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia

Abstract

We show that piecewise smooth maps with a finite number of pieces of monotonicity and nowhere vanishing Lipschitz continuous derivative can have only period doubling solenoids. The proof is based on the fact that if

p_{1} < ... < p_{n}

is a periodic orbit of a continuous map f then there is a union set

{q_{1}, ..., q_{n - 1}}

of some periodic orbits of f such that

p_{i} < q_{i} < p_{i + 1}

for any i.

Keywords

ENG

Markov graph, periodic point, piecewise smooth map with nowhere vanishing Lipschitz continuous derivative, piecewise linear map, solenoid

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Title

All solenoids of piecewise smooth maps are period doubling

Affiliations

Abstract

Keywords

Bibliography