$C^1$-robust homoclinic tangencies

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}.