American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2006, 16 (2) : 463-504. doi: 10.3934/dcds.2006.16.463
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