Cyclic renormalization and automorphism groups of rooted trees
type: Item
instance of: scholarly article
title: Cyclic renormalization and automorphism groups of rooted trees
zbMATH Open document ID: 0847.58019
author: Hyman Bass
author: Maria Victoria Otero-Espinar
author: Daniel N. Rockmore
author: Charles Tresser
published in: Lecture Notes in Mathematics
publication date: 1996-01-11T00:00:00Z
review text: To quote from the Introduction, the book is motivated by the ``relation between renormalizability of certain dynamical systems on the unit interval and group actions on rooted trees. Certain classes of maps of the unit interval, when restricted to invariant Cantor sets, have the form of such automorphisms and better understanding the structure of such automorphism groups contributes to a fuller understanding of the types of maps which have these sorts of restrictions''. There are four main chapters, the first of which gives definitions and develops background on renormalization, minimal sets, order topology, and interval renormalization. Particular attention is paid to the case of infinite renormalizability. Differences between dynamics arising from minimal sets in the interval and in the circle are highlighted. Chapter II discusses itineraries of orbits of unimodal interval maps and describes renormalization of the dynamical system obtained by restricting the map to an orbit closure and factorization in the space of possible itineraries. Other relevant aspects of the theory of interval maps are discussed. The last two chapters deal mostly with group theory. They are motivated by the dynamical considerations of the first two chapters, but are for the most part independent of them. Chapter III deals with ``spherically transitive automorphisms of rooted trees'', where a rooted tree is an acyclic, locally finite, connected graph with a distinguished root vertex. Each vertex on such a tree has a magnitude, which is the minimal number of edges of any path from the vertex to the root. The \(n\)-sphere \(X_n\) of the tree is the set of all vertices of magnitude \(n\); each vertex of \(X_{n + 1}\) connects to a single vertex of \(X_n\). The tree is `spherical' if for each \(n\) the number of vertices of \(X_{n + 1}\) attached to a vertex \(x \in X_n\) is independent of the choice of \(x \in X_n\). Any automorphism of a rooted tree must send the root to itself, and hence mus…
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 37-02
zbMATH DE Number: 833159
zbMATH Keywords: renormalizability
zbMATH Keywords: dynamical systems
zbMATH Keywords: unit interval
zbMATH Keywords: group actions
zbMATH Keywords: rooted trees
Wikidata QID: Q56269701
reviewed by: Mike Hurley
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0096321
OpenAlex ID: W319884268
DOI: 10.1007/BFB0096321
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description: scientific article; zbMATH DE number 833159