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Iterated images and the plane Jacobian conjecture
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 117943651, United States 
$ s (2x+1)= \frac{s (x)} {s (2x)}$ $1+\frac{1}{ s (2x1)}1. $
We also present a onetoone 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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