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On hyperbolic measures and periodic orbits
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 |
$ 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 $\alpha > 1$.
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