Abstract
We study hyperbolic maps depending on a parameter ε. Each of them has an invariant Cantor set. As ε tends to zero, the map approaches the boundary of hyperbolicity. We analyze the asymptotics of scaling function of the invariant Cantor set as ε goes to zero. We show that there is a limiting scaling function of the limiting map and this scaling function has dense jump discontinuities because the limiting map is not expanding. Removing these discontinuities by continuous extension, we show that we obtain the scaling function of the limiting map with respect to a Ulam-von Neumann type metric.
Similar content being viewed by others
References
Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Berlin, Heidelberg, New York: Springer
Jiang, Y.: Ratio geometry in dynamical systems. Preprint, June, 1989, IHES
Jiang, Y.: Generalized Ulam-von Neumann transformations. Thesis, Graduate School of CUNY, 1990 and UMI dissertation information service
Sullivan, D.: Differentiable Structure on Fractal Like Sets Determined by Intrinsic Scaling Functions on Dual Cantor Sets. The Proceedings of Symposia in Pure Mathematics, Vol.48, 1988
Thurston, W.: The geometry and topology of three-manifolds. Preprint, Princeton University 1982
Ulam, S.M., von Neumann, J.: On the Combinations of Stochastic and Deterministic Processes. Bull. Am. Math. Soc.53, 1120 (1947)
Author information
Authors and Affiliations
Additional information
Communicated by J.-P. Eckmann
Rights and permissions
About this article
Cite this article
Jiang, Y. Asymptotic differentiable structure on Cantor set. Commun.Math. Phys. 155, 503–509 (1993). https://doi.org/10.1007/BF02096725
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02096725