Dynamics of vertical delay endomorphisms

Abstract

A vertical delay endomorphism F on R k , with k ≥ 2, is the endomorphism associated to the difference equationwhere 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 F of one-parameter families of vertical delay endomorphisms. We will prove that for any {F µ}µ∈R in F and every |µ| large enough, the nonwandering set Ω(Fµ) of Fµ, is either the empty set or an expanding Cantor set and the restriction of Fµ to Ω(Fµ) is conjugated to the unilateral shift on two symbols.

Perturbations of the quadratic family of order two

“…Recall that v ± = (1, t ± ) denotes eigenvectors associated with the eigenvalues λ ± of DG(0). Following the steps of the proof of proposition 1 in [4], one can verify that there exists an α > 0 such that the function W (x, y) = y − t − x − αx 2 satisfies W (G(x, y)) − W (x, y) < 0 for every (x, y) such that W (x, y) < 0.…”