A generic-dimensional property of the invariant measures for circle diffeomorphisms
Shigenori Matsumoto
Journal of Modern Dynamics 2013, 7(4): 553-563 doi: 10.3934/jmd.2013.7.553
Given any Liouville number $\alpha$, it is shown that the nullity of the Hausdorff dimension of the invariant measure is generic in the space of the orientation-preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.
keywords: Hausdorff dimension fast approximation by conjugation. invariant measure rotation number Liouville number Circle diffeomorphism
Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem
Matilde Martínez Shigenori Matsumoto Alberto Verjovsky
Journal of Modern Dynamics 2016, 10(02): 113-134 doi: 10.3934/jmd.2016.10.113
We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathfrak{F})$ of a compact minimal lamination $(M,\mathfrak{F})$ 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 $\mathfrak{F}$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.
keywords: Hyperbolic surfaces horocycle and geodesic flows hyperbolic laminations minimality.

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