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Robust Heterodimensional Cycles and Tame Dynamics

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New Trends in Mathematical Physics

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Abstract

A diffeomorphism f has a heterodimensional cycle if there are hyperbolic sets Λ and Σ with different indices (dimension of the unstable bundle) such that the unstable manifold of Λ meets the stable one of Σ and vice-versa. This cycle has co-index one if index(Λ)=index(Σ)±1 and is robust if, for every g close to f, the continuations of Λ and Σ for g have a cycle. In the C 1-setting, we discuss the generation of robust heterodimensional cycles by heterodimensional ones and obtain some consequences of this phenomenon for tame dynamics.

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Authors and Affiliations

  1. Departamento de Matemática, PUC-Rio, Marquês de São Vicente 225, Gávea, Rio de Janeiro, 225453-900, Brazil

    Lorenzo J. Díaz

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  1. Lorenzo J. Díaz
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Correspondence to Lorenzo J. Díaz .

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  1. Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, RJ, Brazil

    Vladas Sidoravičius

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Dedicated to Paul Schweitzer (now 70 years young)

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Díaz, L.J. (2009). Robust Heterodimensional Cycles and Tame Dynamics. In: Sidoravičius, V. (eds) New Trends in Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2810-5_17

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