Locally topologically generic diffeomorphisms with Lyapunov unstable\n Milnor attractors

2016/04/08 by Ivan Shilin, Shilin, Ivan
Biochemistry, Genetics and Molecular Biology · Mathematics · #37B25 (Primary) 37B20 #37C20 #37C29 #37D30 (Secondary) #Caveolin-1 and cellular processes #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1604.02437

openalex publication_date 2016/04/08 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We prove that for every smooth compact manifold M and any r \≥ 1,\nwhenever there is an open domain in \Diff^r(M) exhibiting a\npersistent homoclinic tangency related to a basic set with a sectionally\ndissipative periodic saddle, topologically generic diffeomorphisms in this\ndomain have Lyapunov unstable Milnor attractors. This implies, in particular,\nthat the instability of Milnor attractors is locally topologically generic in\nC¹ if dim ,M \≥ 3 and in C² if dim ,M = 2.\nMoreover, it follows from the results of C. Bonatti, L. J. D 'iaz and E. R.\nPujals that, for a C¹ topologically generic diffeomorphism of a closed\nmanifold, either any homoclinic class admits some dominated splitting, or this\ndiffeomorphism has an unstable Milnor attractor, or the inverse diffeomorphism\nhas an unstable Milnor attractor. The same results hold for statistical and\nminimal attractors.\n

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