C¹-robust homoclinic tangencies

2024/06/18 by Dongchen Li, Li, Dongchen
Mathematics · #37C05 #37C29 #37C75 #37D25 #Advanced Banach Space Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Fixed Point Theorems Analysis #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2406.12500

openalex publication_date 2024/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 C¹-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 C¹-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 Díaz in \citepBonDia:12b.

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