Asymptotic geometry of hyperbolic well-ordered Cantor sets. (English) Zbl 1083.37509
Summary: We study the well-ordered Cantor sets in hyperbolic sets on the line and the plane. Examples of such sets occur in circle maps and in area-preserving twist maps. We set up a renormalization scheme employing in both cases the first return map. We prove convergence of this scheme. The convergence implies that the asymptotic geometry of such a well-ordered set with irrational rotation number and their nearby well-ordered orbits is determined by the Lyapunov exponent of this set.
MSC:
| 37D05 | Dynamical systems with hyperbolic orbits and sets |
Keywords:
renormalization; hyperbolic Cantor sets; Lyapunov exponents; bounded nonlinearity; Denjoy-Koksma; Aubry-Mather sets; asymptotic geometryReferences:
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