Dynamics of vertical delay endomorphisms

N. Romero; A. Rovella; F. Vilamajó

doi:10.3934/dcdsb.2003.3.409

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2003, Volume 3, Issue 3: 409-422. Doi: 10.3934/dcdsb.2003.3.409

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Dynamics of vertical delay endomorphisms

1.
Universidad Centro Occidental Lisandro Alvarado, Decanato de Ciencias y Tecnología, Departamento de Matemática, Apartado Postal 400, Barquisimeto
2.
Universidad de la República, Facultad de Ciencias, Centro de Matemática, Igua 425 C.P. 11400 Montevideo
3.
Universidad Politècnica de Catalunya, Departament de Matemàtica Aplicada 2, Escola Tècnica Superior D'Enginyeria Industrial, Colom 11, 08222 Terrassa, Barcelona

Received: September 30, 2002

Revised: December 31, 2002

Published: July 2003

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Abstract

A vertical delay endomorphism $F$ on $\mathbb{R}^k$, with $k\ge 2$, is the endomorphism associated to the difference equation $x_{n+k}=f(x_n,\cdots,x_{n+k-1})$, where the function $f$ is $C^2$ and its partial derivative of second order with respect to the first variable is bigger than every other partial derivative of second order. The main goal of this paper is to describe the dynamical behaviour of a huge class $\mathcal{F}$ of one-parameter families of vertical delay endomorphisms. We will prove that for any $\{F_\mu\}_{\mu\in\mathbb{R}}$ in $\mathcal{F}$ and every $|\mu|$ large enough, the nonwandering set $\Omega(F_\mu)$ of $F_\mu$, is either the empty set or an expanding Cantor set and the restriction of $F_{\mu}$ to $\Omega(F_\mu)$ is conjugated to the unilateral shift on two symbols.

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Mathematics Subject Classification: 37C05, 37D20.

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