The circle and the solenoid

• Type: Article in International Scientific Journal
• Date: 2006

Authors

Pinto, AA (Author) FCUP; Sullivan, D (Author) Other

Journal

Title: Discrete and Continuous Dynamical Systems

Vol. 16 No. 2

Pages: 463-504

ISSN: 1078-0947

Publisher: American Institute of Mathematical Sciences

Indexing

ISI Web of Knowledge - 11 Citations

Scopus - 13 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-004-GRS

Resumo (PT): 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 .

Abstract (EN): 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.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 42