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Frontiers in Complex DynamicsIn Celebration of John Milnor's 80th Birthday$
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Araceli Bonifant, Misha Lyubich, and Scott Sutherland

Print publication date: 2014

Print ISBN-13: 9780691159294

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159294.001.0001

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Metric stability for random walks (with applications in renormalization theory)

Metric stability for random walks (with applications in renormalization theory)

Chapter:
(p.261) Metric stability for random walks (with applications in renormalization theory)
Source:
Frontiers in Complex Dynamics
Author(s):

Carlos Gustavo Moreira

Daniel Smania

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159294.003.0013

This chapter considers the stability of metric (measure-theoretic) properties of dynamical systems. A well-known example is that of (C²) expanding maps on the circle; this class is structurally stable, and all such maps have an absolutely continuous and ergodic invariant probability satisfying certain decay of correlations estimates. In particular, in the measure theoretic sense, most of the orbits are dense in the phase space. The chapter uses the idea of random walk, which describes transitions between various dynamical scales, to prove a surprising rigidity result: the conjugacy between two unimodal maps of the same degree with Feigenbaum or wild attractors is absolutely continuous.

Keywords:   random walks, metric stability, robustness, renormalization theory, dynamical scales, dynamical systems, rigidity, conjugacy

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