## Institute for Mathematical Sciences

## Preprint ims04-06

** A. A. Pinto and D. Sullivan**
* Dynamical Systems Applied to Asymptotic Geometry*

Abstract: In the paper we discuss two questions about smooth expanding dynamical systems on the circle. (i) We characterize the sequences
of asymptotic length ratios which occur for systems with H\"older
continuous derivative. The sequence of asymptotic length ratios are
precisely those given by a positive H\"older continuous function $s$
(solenoid function) on the Cantor set $C$ of $2$-adic integers
satisfying a functional equation called the matching condition. The
functional equation for the $2$-adic integer Cantor set is $$ s
(2x+1)= \frac{s (x)} {s (2x)}
\left( 1+\frac{1}{ s (2x-1)}\right)-1.
$$ We also present a one-to-one correspondence between solenoid
functions and affine classes of $2$-adic quasiperiodic tilings of
the real line that are fixed points of the 2-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
$\alpha$-H\"older continuous. The maximum smoothness is
$C^{2+\alpha}$ for $0 < \alpha \le 1$ if, and only if, $cr$ is
$(1+\alpha)$-H\"older. A curious connection with Mostow type
rigidity is provided by the fact that $s$ must be constant if it
is $\alpha$-H\"older for $\alpha > 1$.

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