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Register a new userHorocycle flows without minimal sets
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Authors
Shigenori Matsumoto
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We show that the horocycle flows of open tight hyperbolic surfaces do not admit minimal sets.
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We show that the horocycle flow associated with a foliation on a compact manifold by hyperbolic surfaces is minimal under certain conditions.
We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(hat M, T^1mathcal{F})$ of a compact minimal lamination $(M,mathcal F)$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal, and examples where this action is not minimal. In the first case, we prove that if $mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.
Theorem 2 of A. Kercheval, Denjoy minimal sets are far from affine, Ergodic Theory and Dynamical Systems 22 (2002), 1803-1812 is corrected by adding a C^2 bound to the hypotheses.
We study quasiperiodically forced circle endomorphisms, homotopic to the identity, and show that under suitable conditions these exhibit uncountably many minimal sets with a complicated structure, to which we refer to as `strangely dispersed. Along the way, we generalise some well-known results about circle endomorphisms to the uniquely ergodically forced case. Namely, all rotation numbers in the rotation interval of a uniquely ergodically forced circle endomorphism are realised on minimal sets, and if the rotation interval has non-empty interior then the topological entropy is strictly positive. The results apply in particular to the quasiperiodically forced Arnold circle map, which serves as a paradigm example.
We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel-Anosov-Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorphism of a new space that admits a homeomorphism that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms of intermediate complexity. Additionally, we show that such examples occur as minimal sets of volume preserving smooth diffeomorphisms of 4-dimensional manifolds. We also use our techniques to exhibit the first examples of minimal, uniformly rigid and weakly mixing homeomorphisms in dimension $1$, and these can also be realized as invariant sets of smooth diffeomorphisms of a 4-manifold. Until now the only known examples of spaces that admit minimal, uniformly rigid and weakly mixing homeomorphisms were modifications of those given by Glasner and Maon in dimension at least $2$.
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