29 research outputs found
Geometry and entropies in a fixed conformal class on surfaces
We show the flexibility of the metric entropy and obtain additional
restrictions on the topological entropy of geodesic flow on closed surfaces of
negative Euler characteristic with smooth non-positively curved Riemannian
metrics with fixed total area in a fixed conformal class. Moreover, we obtain a
collar lemma, a thick-thin decomposition, and precompactness for the considered
class of metrics. Also, we extend some of the results to metrics of fixed total
area in a fixed conformal class with no focal points and with some integral
bounds on the positive part of the Gaussian curvature.Comment: Minor changes to exposition. Final version, to appear in Annales de
l'Institut Fourie
Knot theory of R-covered Anosov flows: homotopy versus isotopy of closed orbits
In this article, we study the knots realized by periodic orbits of R-covered
Anosov flows in compact 3-manifolds. We show that if two orbits are freely
homotopic then in fact they are isotopic. We show that lifts of periodic orbits
to the universal cover are unknotted. When the manifold is atoroidal, we deduce
some finer properties regarding the existence of embedded cylinders connecting
two given homotopic orbits.Comment: 20 pages, 9 figure
Entropy rigidity of Hilbert and Riemannian metrics
In this paper we provide two new characterizations of real hyperbolic
-space using the Poincar\'e exponent of a discrete group and the volume
growth entropy. The first characterization is in the space of Hilbert metrics
and generalizes a result of Crampon. The second is in the space of Riemannian
metrics with Ricci curvature bounded below and generalizes a result of
Ledrappier and Wang.Comment: 14 pages, some revisions following the referees remarks. To be
published in IMR