Smooth classification of geometrically finite one-dimensional maps

Jiang, Yunping

doi:10.1090/S0002-9947-96-01487-0

Smooth classification of geometrically finite one-dimensional maps
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Trans. Amer. Math. Soc. 348 (1996), 2391-2412 Request permission

Abstract:

The scaling function of a one-dimensional Markov map is defined and studied. We prove that the scaling function of a non-critical geometrically finite one-dimensional map is Hölder continuous, while the scaling function of a critical geometrically finite one-dimensional map is discontinuous. We prove that scaling functions determine Lipschitz conjugacy classes, and moreover, that the scaling function and the exponents and asymmetries of a geometrically finite one-dimensional map are complete $C^{1}$-invariants within a mixing topological conjugacy class.

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