Dynamical Systems Commons^™

Dynamics Of The Meromorphic Families $F_\Lambda=\Lambda \Tan^Pz^Q$, Tao Chen, Linda Keen

Publications and Research

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials P_a(z) =z^d−1(z−da/(d−1)), the family f_λ=λtan^pz^q. These functions have a super-attractive fixed point, and, depending on p, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard …

Iterated Aluthge Transforms: A Brief Survey, Jorge Antezana, Enrique R. Pujals, Demetrio Stojanoff

Publications and Research

Given an r × r complex matrix T, if T = U|T| is the polar decomposition of T, then the Aluthge transform is defined by

∆(T) = |T|^1/2U|T|^1/2.

Let ∆ⁿ(T) denote the n-times iterated Aluthge transform of T, i.e. ∆⁰(T) = T and ∆ⁿ(T) = ∆(∆ⁿ⁻¹(T)), n ∈ N. In this paper we make a brief survey on the known properties and applications of …

A Remark On Conservative Diffeomorphisms, Jairo Bochi, Bassam R. Fayad, Enrique Pujals

Publications and Research

Abstract:

We show that a stably ergodic diffeomorphism can be C¹ approximated by a diffeomorphism having stably non-zero Lyapunov exponents.

Résumé:

On montre qu'un difféomorphisme stablement ergodique peut être C¹ approché par un difféomorphisme ayant des exposants de Lyapunov stablement non-nuls.

Robustly Transitive Sets And Heterodimensional Cycles, Christian Bonatti, Lorenzo J. Díaz, Enrique R. Pujals, Jorge Rocha

Publications and Research

It is known that all non-hyperbolic robustly transitive sets Λ_φ have a dominated splitting and, generically, contain periodic points of different indices. We show that, for a C¹-dense open subset of diffeomorphisms φ, the indices of periodic points in a robust transitive set Λ_φ form an interval in ℕ. We also prove that the homoclinic classes of two periodic points in Λ_φ are robustly equal. Finally, we describe what sort of homoclinic tangencies may appear in Λ_φ by studying its dominated splittings.