Hostname: page-component-cb9f654ff-65tv2 Total loading time: 0 Render date: 2025-08-20T03:19:55.348Z Has data issue: false hasContentIssue false

Ratio geometry, rigidity and the scenery process forhyperbolic Cantor sets

Published online by Cambridge University Press:  01 June 1997

TIM BEDFORD
Affiliation:
Faculty of Mathematics, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands
ALBERT M. FISHER
Affiliation:
SUNY at Stony Brook, USA

Abstract

Given a ${\cal C}^{1+\gamma}$ hyperbolic Cantor set$C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down towarda point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to anergodic Cantor set valued process. The values of this process, calledlimit sets, are indexed by a Hölder continuous set-valuedfunction definedon Sullivan's dual Cantor set. We show the limit sets are themselves ${\calC}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets,with the highest degree of smoothness which occurs in the ${\calC}^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to thefollowing rigidity theorem: if two ${\cal C}^{k+\gamma},{\cal C}^\infty$ or${\cal C}^\omega$ hyperbolic Cantor sets are ${\cal C}^1$ conjugate, then theconjugacy (with a different extension) is in fact already ${\calC}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$. Within one ${\calC}^{1+\gamma}$ conjugacy class, each smoothness class is a Banach manifold,which is acted on by the semigroup given by rescaling subintervals.Smoothness classes nest down, and contained in the intersection of them allis a compact set which is the attractor for the semigroup: the collection oflimit sets. Convergence is exponentially fast, in the ${\cal C}^1$ norm.

Information

Type
Research Article
Copyright
1997 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable