Abstract
We prove that a $C^{1}$ orientation-preserving circle endomorphism which is Hölder conjugate to a $C^{1}$ circle expanding endomorphism is itself expanding.
Highlights
Introduction and the main theoremSuppose M is a C2 dimension n ≥ 1 Riemanian manifold
F is β-Holder conjugate to g, we mean that there is a homeomorphism h of the unit circle such that f ◦h=h◦g and h is β-Holder continuous. (We do not need any additional condition on h−1.) We would like to note that Holdericity is used only for the exponential decay property for Markov partitions from g to f
The theorem is true for any one-dimensional Markov map as follows
Summary
Introduction and the main theoremSuppose M is a C2 dimension n ≥ 1 Riemanian manifold. Suppose f is β-Holder conjugate to a C1 expanding endomorphism (or a C1 Anosov diffeomorphism) for some 0 < β ≤ 1. Suppose f is β-Holder conjugate to a C1 expanding endomorphism g : M → M for some 0 < β ≤ 1.
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