Periodic points on the $2$-sphere
Charles Pugh Michael Shub
Discrete & Continuous Dynamical Systems - A 2014, 34(3): 1171-1182 doi: 10.3934/dcds.2014.34.1171
For a $C^{1}$ degree two latitude preserving endomorphism $f$ of the $2$-sphere, we show that for each $n$, $f$ has at least $2^{n}$ periodic points of period $n$.
keywords: latitude preserving. Periodic points smoothness two sphere degree two
Hölder foliations, revisited
Charles Pugh Michael Shub Amie Wilkinson
Journal of Modern Dynamics 2012, 6(1): 79-120 doi: 10.3934/jmd.2012.6.79
We investigate transverse Hölder regularity of some canonical leaf conjugacies in normally hyperbolic dynamical systems and transverse Hölder regularity of some invariant foliations. Our results validate claims made elsewhere in the literature.
keywords: normal hyperbolicity invariant foliations H\"older regularity of foliations. Partial hyperbolicity
Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms
Charles Pugh Michael Shub Alexander Starkov
Discrete & Continuous Dynamical Systems - A 2006, 14(4): 845-855 doi: 10.3934/dcds.2006.14.845
In 1954, F. Mautner gave a simple representation theoretic argument that for compact surfaces of constant negative curvature, invariance of a function along the geodesic flow implies invariance along the horocycle flows (these are facts which imply ergodicity of the geodesic flow itself), [M]. Many generalizations of this Mautner phenomenon exist in representation theory, [St1]. Here, we establish a new generalization, Theorem 2.1, whose novelty is mostly its method of proof, namely the Anosov-Hopf ergodicity argument from dynamical systems. Using some structural properties of Lie groups, we also show that stable ergodicity is equivalent to the unique ergodicity of the strong stable manifold foliations in the context of affine diffeomorphisms.
keywords: manifold foliations affine diffeomorphisms. stable ergodicity Representation theory the Anosov-Hopf ergodicity Lie groups unique ergodicity

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