Please use this identifier to cite or link to this item: https://hdl.handle.net/10216/100677
Author(s): Pinto, AA
Sullivan, D
Title: The circle and the solenoid
Issue Date: 2006
Abstract: In the paper, we discuss two questions about degree d smooth expanding circle maps, with d >= 2. (i) We characterize the sequences of asymptotic length ratios which occur for systems with Holder continuous derivative. The sequences of asymptotic length ratios are precisely those given by a positive Holder continuous function s (solenoid function) on the Cantor set C of d-adic integers satisfying a functional equation called the matching condition. In the case of the 2-adic integer Cantor set, the functional equation is s(2x + 1) = s(x)/s(2x) 1 + 1s(2x-1) -1. We also present a one-to-one correspondence between solenoid functions and affine classes of exponentially fast d-adic tilings of the real line that are fixed points of the d-amalgamation operator. (ii) We calculate the precise maximum possible level of smoothness for a representative of the system, up to diffeomorphic conjugacy, in terms of the functions s and cr(x) = (1 + s(x))/(1 + (s(x + 1))(-1)). For example, in the Lipschitz structure on C determined by s, the maximum smoothness is C1+alpha for 0 < alpha <= 1 if and only if s is alpha-Holder continuous. The maximum smoothness is C2+alpha for 0 < alpha <= 1 if and only if cr is (1 + alpha)-Holder. A curious connection with Mostow type rigidity is provided by the fact that s must be constant if it is alpha-Holder for alpha > 1.
Description: In the paper, we discuss two questions about degree smooth expanding circle maps, with . (i) We characterize the sequences of asymptotic length ratios which occur for systems with Hölder continuous derivative. The sequence of asymptotic length ratios are precisely those given by a positive Hölder continuous function (solenoid function) on the Cantor set of -adic integers satisfying a functional equation called the matching condition. In the case of the -adic integer Cantor set, the functional equation is We also present a one-to-one correspondence between solenoid functions and affine classes of exponentially fast -adic tilings of the real line that are fixed points of the -amalgamation operator. (ii) We calculate the precise maximum possible level of smoothness for a representative of the system, up to diffeomorphic conjugacy, in terms of the functions and . For example, in the Lipschitz structure on determined by , the maximum smoothness is for if and only if is -Hölder continuous. The maximum smoothness is for if and only if is -Hölder. A curious connection with Mostow type rigidity is provided by the fact that must be constant if it is -Hölder for .
Subject: Matemática
Mathematics
Scientific areas: Ciências exactas e naturais::Matemática
Natural sciences::Mathematics
URI: https://hdl.handle.net/10216/100677
Document Type: Artigo em Revista Científica Internacional
Rights: restrictedAccess
Appears in Collections:FCUP - Artigo em Revista Científica Internacional

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