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Transversality in the setting of hyperbolic and parabolic maps

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Abstract

In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in [24] to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcations at parabolic periodic points are generic, or one has persistency of periodic orbits with a fixed multiplier.

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University in Jerusalem, Givat Ram, Jerusalm, 91904, Israel

    Genadi Levin

  2. Shanghai Center for Mathematical Sciences, Fudan University, 2005 Songhu Road, Shanghai, 200438, China

    Weixiao Shen

  3. Department of Mathematics, Imperial College London, 180 Queen’s Gate, SW7 2AZ, London, UK

    Sebastian van Strien

Authors
  1. Genadi Levin
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  2. Weixiao Shen
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  3. Sebastian van Strien
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Correspondence to Genadi Levin.

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Dedicated to Lawrence Zalcman

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Levin, G., Shen, W. & van Strien, S. Transversality in the setting of hyperbolic and parabolic maps. JAMA 141, 247–284 (2020). https://doi.org/10.1007/s11854-020-0130-7

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  • DOI: https://doi.org/10.1007/s11854-020-0130-7

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