Cyclic renormalization and automorphism groups of rooted trees
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 must preserve each \(n\)-sphere. Any automorphism of a spherical tree is `spherically transitive' if its restriction to any \(n\)-sphere is a cyclic permutation. The structures of the set of these automorphisms, and of the set of all automorphisms, are described in terms of wreath products. The final chapter makes use of this wreath product structure to describe the closed normal subgroups of the automorphism group of a spherical rooted tree.
scientific article; zbMATH DE number 833159