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DCDS

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 $n$.

JMD

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.

DCDS

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.

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