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Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics

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Abstract

We consider semigroup actions on the unit interval generated by strictly increasing \(C^r\)-maps. We assume that one of the generators has a pair of fixed points, one attracting and one repelling, and a heteroclinic orbit that connects the repeller and attractor. We also assume that the other generators form a robust blender, which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller. This is a model setting for partially hyperbolic systems with one central direction. We show that, under additional conditions on \(\frac{f''}{f'}\) and the Schwarzian derivative, the above semigroups exhibit, \(C^r\)-generically for any \(r \ge 3\), arbitrarily fast growth of the number of periodic points as a function of the period. We also show that a \(C^r\)-generic semigroup from the class under consideration supports an ultimately complicated behavior called universal dynamics.

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Notes

  1. If \(\tau _S(z_1,f_0)\ne \tau _S(z_2)\), then choose \((i_1,i_2)=(1,2)\). If \(\tau _S(z_1,f_0)= \tau _S(z_2)=+1\) and \(\tau _A(z_4,f_0)=+1\), then choose \((i_1,i_2)=(2,4)\). Other cases are similar.

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Acknowledgments

The authors thank to Pierre Berger for useful dicsussions. KS and DT are grateful for the hospitality of Department of Mathematics of Kyoto University. This paper is supported by GCOE program of Kyoto University, JSPS KAKENHI Grant-in-Aid for Young Scientists (A) (22684003), Scientific Research (C) (26400085), and JSPS Fellows (26\(\cdot \)1121), by Grant No. 14-41-00044 of RSF (Russia), and by the Royal Society Grant IE141468.

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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Kyoto, 606-8502, Japan

    Masayuki Asaoka

  2. Graduate School of Commerce and Management, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, 186-8601, Japan

    Katsutoshi Shinohara

  3. Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, UK

    Dmitry Turaev

  4. Lobachevsky University of Nizhny Novgorod, Nizhny Novgorod, 603950, Russia

    Dmitry Turaev

Authors
  1. Masayuki Asaoka
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  2. Katsutoshi Shinohara
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  3. Dmitry Turaev
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Correspondence to Masayuki Asaoka.

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Communicated by Nalini Anantharaman.

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Asaoka, M., Shinohara, K. & Turaev, D. Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics. Math. Ann. 368, 1277–1309 (2017). https://doi.org/10.1007/s00208-016-1468-0

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  • DOI: https://doi.org/10.1007/s00208-016-1468-0

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