$C^1$-robust homoclinic tangencies

TL;DR

This paper introduces standard blenders and demonstrates their role in generating $C^1$-robust homoclinic tangencies, especially after perturbations of diffeomorphisms with heterodimensional cycles, and explores their implications for dynamical systems.

Contribution

It establishes the fundamental properties of standard blenders and shows their application in creating uncountably many $C^1$-robust homoclinic tangencies under certain conditions.

Findings

Blenders appear after small perturbations of diffeomorphisms with heterodimensional cycles.

Unfolding homoclinic tangencies can produce uncountably many $C^1$-robust tangencies.

The results answer a question posed by Bonatti and Díaz.

Abstract

The aim of this paper is twofold. First, we introduce standard blenders (special hyperbolic sets) and their variations, and prove their fundamental properties on the generation of $C^1$-robust tangencies. In particular, these blenders appear after $C^r$-small perturbations of any diffeomorphism having a heterodimensional cycle of coindex 1. Next, as an application, we show that unfolding a homoclinic tangency to a hyperbolic periodic point can produce uncountably many $C^1$-robust homoclinic tangencies, provided that either this point is involved in a coindex-1 heterodimensional cycle, or the central dynamics near it is not essentially two-dimensional. The result answers a question posed by Bonatti and D{\'i}az in \citep{BonDia:12b}.