June  2006, 16(2): 463-504. doi: 10.3934/dcds.2006.16.463

The circle and the solenoid

1. 

DMP, Faculdade de Ciências, Universidade do Porto, 4000 Porto, Portugal

2. 

Einstein chair, Graduate Center, City University of New York and SUNY Stony Brook, New York 11794-3651, United States

Received  January 2005 Revised  January 2006 Published  July 2006

In the paper, we discuss two questions about degree $d$ smooth expanding circle maps, with $d \ge 2$. (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 $s$ (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)= \frac{s (x)} {s (2x)}$ $1+\frac{1}{ s (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 $C^{1+\alpha}$ for $0 < \alpha \le 1$ if and only if $s$ is $\alpha$-Hölder continuous. The maximum smoothness is $C^{2+\alpha}$ for $0 < \alpha \le 1$ if and only if $cr$ is $(1+\alpha)$-Hölder. A curious connection with Mostow type rigidity is provided by the fact that $s$ must be constant if it is $\alpha$-Hölder for $\alpha > 1$.

Citation: A. A. Pinto, D. Sullivan. The circle and the solenoid. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 463-504. doi: 10.3934/dcds.2006.16.463
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