We show a Cr connecting lemma for area-preserving surface diffeomorphisms and for periodic Hamiltonian on surfaces. We prove that for a generic Cr , r = 1, 2, . . ., ∞, area-preserving diffeomorphism on a compact orientable surface, homotopic to identity, every hyperbolic periodic point has a transversal homoclinic point. We also show that for a Cr, r = 1, 2, . . ., ∞ generic time periodic Hami...

We examine the phase space structures that govern reaction dynamics in the absence of critical points on the potential energy surface. We show that in the vicinity of hyperbolic invariant tori, it is possible to define phase space dividing surfaces that are analogous to the dividing surfaces governing transition from reactants to products near a critical point of the potential energy surface. W...

We consider the problem of determining the conformai type—hyperbolic or parabolic—of a covering surface of the Riemann sphere with n punctures. To such a surface there corresponds a graph called the Speiser graph of the covering, and it is natural to ask for a criterion for the type of the surface in terms of properties of the graph. We show how to define a random walk on the vertices of the gr...

The end of a hyperbolic surface is studied in terms of the behavior at innnity of geodesics on the surface. For a class of surfaces called untwisted utes it is possible to give a fairly precise description of the ending geometry. From the point of view of a Fuchsian group representing such a surface this provides new information about the existence of Dirichlet and Garnett points. 0. Introducti...

Consider a simply connected Riemann surface represented by a Speiser graph. Nevanlinna asked if the type of the surface is determined by the mean excess of the graph: whether mean excess zero implies that the surface is parabolic and negative mean excess implies that the surface is hyperbolic. Teichmüller gave an example of a hyperbolic simply connected Riemann surface whose mean excess is zero...

The space of broken hyperbolic structures generalizes the usual Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations–or equivalently, the space of projectivized affine foliations of a punctured surface–likewise admits a generalization to projectivized broken measured foliations. Just as projectivized measured foliations provide Thurston's boundary ...

We consider dynamical systems generated by partially hyperbolic surface endomorphisms of class Cr with one-dimensional strongly unstable subbundle. As the main result, we prove that such a dynamical system generically admits finitely many ergodic physical measures whose union of basins of attraction has total Lebesgue measure, provided that r ≥ 19.

In [Z1] S. Zelditch introduced an equivariant version of a pseudo-differential calculus on a hyperbolic Riemann surface. We recast his construction in terms of trilinear invariant functionals on irreducible unitary representations of PGL2(R). This allows us to use certain properties of these functionals in the study of the action of pseudodifferential operators on eigenfunctions of the Laplacia...

We consider properly immersed finite topology minimal surfaces Σ in complete finite volume hyperbolic 3-manifolds N , and in M × S, where M is a complete hyperbolic surface of finite area. We prove Σ has finite total curvature equal to 2π times the Euler characteristic χ(Σ) of Σ, and we describe the geometry of the ends of Σ. .

Cooper and Long generalised Epstein and Penner’s Euclidean cell decomposition of cusped hyperbolic n–manifolds of finite volume to non-compact strictly convex projective n–manifolds of finite volume. We show that Weeks’ algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projective surfaces.