Dynamics on the unit disk: Short geodesics and simple cycles

Curtis T Mcmullen

doi:10.4171/cmh/209

Abstract

Dynamics on the unit disk: Short geodesics and simple cycles

Highlights

CMH that varies continuously with f, conjugates f to pd .z/ D zd, and satisfies f .z/ D z when f D pd
The degree of a cycle C is the least e > 0 such that pd jC extends to a covering map of the circle of degree e
Any cycle with LS.C; f / < log 2 is simple. All such cyclSes Ci have the same rotation number, and pd j Ci preserves the cyclic ordering of Ci

Summary

Simple cycles

We discuss the combinatorics of periodic cycles for the map pd .t / D d t mod 1, and prove the closure of the simple cycles has Hausdorff dimension zero. We let Cd denote the set of all cycles of degree d , and Cd .p=q/ Cd the simple cycles with rotation number p=q. The fixed-point portrait [Gol] of a simple cycle C 2 Cd .p=q/ is the monotone increasing function. (1) A simple cycle C 2 Cd .p=q/ is uniquely determined by its fixed-point portrait .j /, and all possible monotone increasing functions .j / arise. A simple cycle C 2 Cd can be reconstructed explicitly from its rotation number p=q and its fixed-point portrait as follows. In general there are q C 1 cubic simple cycles with rotation number p=q, whose fixed-point portraits are given by .1/ D 0; 1; : : : ; q. Let Pd .n; p=q/ denote the set of all simple precycles of length n and rotation number p=q.

Blaschke products

Bounds on repelling cycles

Short cycles and short geodesics

D 1 X0 j1 q j

Binding and renormalization