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Dynamics and Universality of Unimodal Mappings with Infinite Criticality

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Abstract

We consider infinitely renormalizable unimodal mappings with topological type which are periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization for finite criticalities converges, uniformly on the real domain, to a mapping of the limiting type.

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Authors and Affiliations

  1. Dept. of Math., Hebrew University, Jerusalem, 91904, Israel

    Genadi Levin

  2. Dept. of Math., Penn State University, University Park, PA, 16802, USA

    Grzegorz Światek

Authors
  1. Genadi Levin
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  2. Grzegorz Światek
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Correspondence to Genadi Levin.

Additional information

Communicated by G. Gallavotti

Both authors were supported by Grant No. 2002062 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.

Partially supported by NSF grant DMS-0245358.

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Levin, G., Światek, G. Dynamics and Universality of Unimodal Mappings with Infinite Criticality. Commun. Math. Phys. 258, 103–133 (2005). https://doi.org/10.1007/s00220-005-1333-6

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  • DOI: https://doi.org/10.1007/s00220-005-1333-6

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