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Multipliers of periodic orbits of quadratic polynomials and the parameter plane

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Abstract

We prove a result about an extension of the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4n/p), and give an explicit condition on internal arguments under which the Julia set of corresponding (unique) infinitely renormalizable quadratic polynomial is not locally connected.

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Authors and Affiliations

  1. Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Genadi Levin

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  1. Genadi Levin
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Correspondence to Genadi Levin.

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In memory of my grandmother Esfir Garbuz

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Levin, G. Multipliers of periodic orbits of quadratic polynomials and the parameter plane. Isr. J. Math. 170, 285–315 (2009). https://doi.org/10.1007/s11856-009-0030-0

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  • DOI: https://doi.org/10.1007/s11856-009-0030-0

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