Global Invertibility of Expanding Maps

In this paper, we show that for every nonnilpotent hyperbolic map $f$ on an infra-nilmanifold, the set $\operatorname{HPer}(f)$ is cofinite in $\mathbb{N}$. This is a generalization of a similar result for expanding maps in Lee and Zhao (J. Math. Soc. Japan 59(1) (2007), 179–184). Moreover, we prove that for every nilpotent map $f$ on an infra-nilmanifold, $\operatorname{HPer}(f)=\{1\}$.

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Expanding endomorphisms of the circle revisited

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570000290x ◽

1985 ◽

Vol 5 (2) ◽

pp. 285-289 ◽

Cited By ~ 44

Author(s):

Michael Shub ◽

Dennis Sullivan

Keyword(s):

Expanding Maps ◽

Tangent Vectors

AbstractTwo Cr, r ≥ 2, expanding maps of the circle which are absolutely continuously conjugate are Cr conjugate. Here ƒ and g: S1 → S1 are expanding if they stretch tangent vectors in some metric, and a conjugacy is an isomorphism h: S1 → S1 such that fh = hg.

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Measures with maximum total exponent and generic properties of $C^{1}$ expanding maps

Hiroshima Mathematical Journal ◽

10.32917/hmj/1389102580 ◽

2013 ◽

Vol 43 (3) ◽

pp. 351-370 ◽

Cited By ~ 3

Author(s):

Takehiko Morita ◽

Yusuke Tokunaga

Keyword(s):

Generic Properties ◽

Expanding Maps

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Coupled-expanding maps under small perturbations

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2011.29.1291 ◽

2011 ◽

Vol 29 (3) ◽

pp. 1291-1307 ◽

Cited By ~ 6

Author(s):

Xu Zhang ◽

◽

Yuming Shi ◽

Guanrong Chen ◽

Keyword(s):

Expanding Maps ◽

Small Perturbations

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Smooth cocycle rigidity for expanding maps, and an application to Mostow rigidity

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004101005710 ◽

2002 ◽

Vol 132 (3) ◽

pp. 439-452 ◽

Cited By ~ 2

Author(s):

OLIVER JENKINSON

Keyword(s):

Lie Group ◽

Piecewise Smooth ◽

Rigidity Theorem ◽

Compact Lie Group ◽

Expanding Maps ◽

Cocycle Rigidity ◽

Mostow Rigidity

We give a variation on the proof of Mostow's rigidity theorem, for certain hyperbolic 3-manifolds. This is based on a rigidity theorem for conjugacies between piecewise-conformal expanding Markov maps. The conjugacy rigidity theorem is deduced from a Livsic cocycle rigidity theorem that we prove for smooth, compact Lie group-valued cocycles over piecewise smooth expanding Markov maps.

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