    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
  Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535  Matthew Nicol. Induced maps of hyperbolic Bernoulli systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 147-154. doi: 10.3934/dcds.2001.7.147  Boris Kalinin, Victoria Sadovskaya. Lyapunov exponents of cocycles over non-uniformly hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5105-5118. doi: 10.3934/dcds.2018224  Nicolai T. A. Haydn, Kasia Wasilewska. Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2585-2611. doi: 10.3934/dcds.2016.36.2585  Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597  Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098  Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099  Sohana Jahan, Hou-Duo Qi. Regularized multidimensional scaling with radial basis functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 543-563. doi: 10.3934/jimo.2016.12.543  Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41  Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75  Khalida Inayat Noor, Muhammad Aslam Noor. Higher order uniformly close-to-convex functions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1277-1290. doi: 10.3934/dcdss.2015.8.1277  F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74  Malo Jézéquel. Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 927-958. doi: 10.3934/dcds.2019039  Jian-Hua Zheng. Dynamics of hyperbolic meromorphic functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2273-2298. doi: 10.3934/dcds.2015.35.2273  Yury Neretin. The group of diffeomorphisms of the circle: Reproducing kernels and analogs of spherical functions. Journal of Geometric Mechanics, 2017, 9 (2) : 207-225. doi: 10.3934/jgm.2017009  Marco Cicalese, Matthias Ruf. Discrete spin systems on random lattices at the bulk scaling. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 101-117. doi: 10.3934/dcdss.2017006  Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007  Marco Lenci. Uniformly expanding Markov maps of the real line: Exactness and infinite mixing. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3867-3903. doi: 10.3934/dcds.2017163  Evelyn Sander. Hyperbolic sets for noninvertible maps and relations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 339-357. doi: 10.3934/dcds.1999.5.339  Vítor Araújo, Ali Tahzibi. Physical measures at the boundary of hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 849-876. doi: 10.3934/dcds.2008.20.849

2019 Impact Factor: 1.338