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Advisor(s)
Abstract(s)
We exhibit the construction of stable arc exchange systems from the stable laminations of hyperbolic diffeomorphisms. We prove a one-to-one correspondence between (i) Lipshitz conjugacy classes of C(1+H) stable arc exchange systems that are C(1+H) fixed points of renormalization and (ii) Lipshitz conjugacy classes of C(1+H) diffeomorphisms f with hyperbolic basic sets Lambda that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Lambda. Let HD(s)(Lambda) and HD(u)(Lambda) be, respectively, the Hausdorff dimension of the stable and unstable leaves intersected with the hyperbolic basic set L. If HD(u)(Lambda) = 1, then the Lipschitz conjugacy is, in fact, a C(1+H) conjugacy in (i) and (ii). We prove that if the stable arc exchange system is a C(1+HDs+alpha) fixed point of renormalization with bounded geometry, then the stable arc exchange system is smooth conjugate to an affine stable arc exchange system.
Description
Keywords
Hyperbolic dynamics Renormalization Markov maps Minimal sets
Citation
A.A. Pinto , D.A. Rand & F. Ferreira (2010) Arc exchange systems and renormalization, Journal of Difference Equations and Applications, 16:4, 347-371, DOI: 10.1080/10236190802422059
Publisher
Taylor & Francis