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Typical coexistence of infinitely many strange attractors

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Abstract

We prove that the coexistence of infinitely many prevalent Hénon-like phenomena is Kolmogorov typical in sectional dissipative \(C^{d,r}\)-Berger domains of parameter families of diffeomorphisms of dimension \(m\ge 3\) for \(d<r-1\). Namely, we answer an old question posed by Colli in [Annales de l’Institut Henri Poincare-Nonlinear Analysis, 15, 539–580 (1998)] on typicality of the coexistence of infinitely many non-hyperbolic strange attractors for \(3\le d<r-1\).

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Notes

  1. The notion of degenerate unfolding is introduced in [4, Def. 3.8] under the name of paratangency. For a more precise definition and contextualization see [8, Sect. 1.4].

  2. The referee pointed out that similarly to Mora-Viana’s proof, Viana’s proof needed one extra assumption on distorsion bound of the determinant of the family. This property seems nonetheless not necessary in view of the alternative proof of [6]

  3. The proof also works for other prevalent phenomenon, such as the existence of a hyperbolic attractor.

  4. Smooth linearization near the saddle is not necessary for this conclusion. See [30] and [14] where the linearizability conditions were removed in the development of the renormalization scheme in the unfolding of generic homoclinic tangencies.

  5. If \(|A\cap B|\ge \Delta |B|\) and \(|J|\ge C |B|\) with \(J\subset B\), then

    $$\begin{aligned} \begin{aligned} |A\cap J|&= { } |(A\cap B)\cap (J\cap B)|=|A\cap B| + |J\cap B| - |(A\cap B)\cup (J\cap B)| \\ {}&{} \ge \Delta |B| + C|B| - |B| =(\Delta +C-1)|B|. \end{aligned} \end{aligned}$$

  6. For every open cover U of a compact metric space X there is a positive real number L, called a Lebesgue number, such that every subset of X of diameter less than L is contained in some element of U.

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Acknowledgements

We thank E. R. Pujals for his guidance, and the encouragement he gave us to write this paper providing many ideas to go ahead. The second author also especially thanks to his supervisor E. R. Pujals for his unconditional friendship and enriching talks on mathematics among other things during his doctorate. Finally, the first author thanks A. Raibekas for his tireless patience and friendship during many difficult moments throughout the process of writing and revising the preliminary versions. The first author was partially supported by CNPq, FAPERJ and grants MTM2017-87697-P and PID2020-113052GB-I00 funded by MCIN (Spain).

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, UFF, Rua Mário Santos Braga s/n - Campus Valonguinhos, Niterói, Brazil

    Pablo G. Barrientos

  2. Universidad del Rosario, Escuela de Ingenieria, Ciencia y Tecnologia, Bogota, Colombia

    Juan David Rojas

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  1. Pablo G. Barrientos
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Correspondence to Pablo G. Barrientos.

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Barrientos, P.G., Rojas, J.D. Typical coexistence of infinitely many strange attractors. Math. Z. 303, 34 (2023). https://doi.org/10.1007/s00209-022-03183-5

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