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Limit drift for complex Feigenbaum mappings

Part of: Complex dynamical systems

Published online by Cambridge University Press:  28 September 2020

GENADI LEVIN  and
GRZEGORZ ŚWIA̧TEK
GENADI LEVIN
Affiliation:
Einstein Institute of Mathematics, Hebrew University, Givat Ram91904, Jerusalem, Israel (e-mail: levin@math.huji.ac.il)
GRZEGORZ ŚWIA̧TEK
Affiliation:
Department of Mathematics and Information Science, Politechnika Warszawska, Koszykowa 75, 00-662, Warszawa, Poland (e-mail: g.swiatek@mini.pw.edu.pl)
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Abstract

We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order $\ell $ of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to $0$ under the dynamics of the tower for corresponding $\ell $ . That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when $\ell $ tends to $\infty $ . We also prove the convergence of the drifts to a finite limit, which can be expressed purely in terms of the limiting tower, which corresponds to a Feigenbaum map with a flat critical point.

Keywords

Feigenbaum functionmeasure of Julia setinvariant measurelimit drift

MSC classification

Primary: 37F25: Renormalization
Secondary: 37F35: Conformal densities and Hausdorff dimension
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 8 , August 2021 , pp. 2428 - 2479
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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