Cyclic Renormalization and Automorphism Groups of Rooted Trees

Hyman Bass, Maria Victoria Otero-Espinar, Daniel Rockmore, Charles Tresser (auth.)9783540605959, 3-540-60595-9

The theme of the monograph is an interplay between dynamical systems and group theory. The authors formalize and study “cyclic renormalization”, a phenomenon which appears naturally for some interval dynamical systems. A possibly infinite hierarchy of such renormalizations is naturally represented by a rooted tree, together with a “spherically transitive” automorphism; the infinite case corresponds to maps with an invariant Cantor set, a class of particular interest for its relevance to the description of the transition to chaos and of the Mandelbrot set. The normal subgroup structure of the automorphism group of such “spherically homogeneous” rooted trees is investigated in some detail. This work will be of interest to researchers in both dynamical systems and group theory.

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Table of contents :
Cyclic renormalization….Pages 1-51
Itinerary calculus and renormalization….Pages 53-101
Spherically transitive automorphisms of rooted trees….Pages 103-133
Closed normal subgroups of Aut(X (q))….Pages 135-156