Ratio geometry, rigidity and the scenery process for hyperbolic Cantor sets

Abstract

Given a C 1+γ hyperbolic Cantor set C, we study the sequence Cn,x of Cantor subsets which nest down toward a point x in C. We show that Cn,x is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on D. Sullivan's dual Cantor set. We show the limit sets are themselves C k+γ , C ∞ or C ω hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the C 1+γ conjugacy class of C. The proof of this leads to the following rigidity theorem: if two C k+γ , C ∞ or C ω hyperbolic Cantor sets are C 1 -conjugate, then the conjugacy (with a different extension) is in fact already C k+γ , C ∞ or C ω . Within one C 1+γ conjugacy class, each smoothness class is a Banach manifold, which is acted on by the semigroup given by rescaling subintervals. Conjugacy classes nest down, and contained in the intersection of them all is a compact set which is the attractor for the semigroup: the collection of limit sets. Convergence is exponentially fast, in the C 1 norm.Typeset by A M S-T E X 1

A family of smooth Cantor sets

“…Then, the result in [Sul88] says that the scaling function is a complete invariant: two C 1+ -regular systems are equivalent if and only if their scaling functions coincide. Moreover, there is a conjugacy which is C 1+ ; for a proof of this see [PT96] and [BF97]. Let us denote by {f p,0 , f p,1 } the IFS which has C p as attractor given by Theorem 1.…”

The Scenery Flow for Geometric Structures on the Torus: The Linear Setting

“…Examples are provided by a generalization of the Ruelle-Perron-Frobenius Theorem to nonstationary subshifts of finite type. This is related to random dynamics on the one hand and to the study of invariant differentiable structures on the other (see [AF00], [BF97] and the references given there).…”

Tangent measure distributions of hyperbolic Cantor sets

“…We first give the definition of a hyperbolic Cantor set (occasionally one encounters slightly different definitions, for a comparison see [6]). …”

“…We shall make use of the following well-known facts about hyperbolic Cantor sets (see [3] or [6] and references therein for proofs). First of all there is the bounded distortion property; There is a constant K>0 such that, for all intervals E C Jxo...x,_l and a, b E E,…”