Diffeomorphisms of the Circle and Hyperbolic Curvature
نویسنده
چکیده
The trace Tf of a smooth function f of a real or complex variable is defined and shown to be invariant under conjugation by Möbius transformations. We associate with a convex curve of class C2 in the unit disk with the Poincaré metric a diffeomorphism of the circle and show that the trace of the diffeomorphism is twice the reciprocal of the geodesic curvature of the curve. Then applying a theorem of Ghys on Schwarzian derivatives we give a new proof of the four-vertex theorem for closed convex curves in the hyperbolic plane.
منابع مشابه
Hyperbolic surfaces of $L_1$-2-type
In this paper, we show that an $L_1$-2-type surface in the three-dimensional hyperbolic space $H^3subset R^4_1$ either is an open piece of a standard Riemannian product $ H^1(-sqrt{1+r^2})times S^{1}(r)$, or it has non constant mean curvature, non constant Gaussian curvature, and non constant principal curvatures.
متن کاملDecay of Correlations for Slowly Mixing Flows
We show that polynomial decay of correlations is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of a class of nonuniformly hyperbolic diffeomorphisms for which Young proved polynomial decay of correlations. Roughly speaking, in situations where the decay rate O(1/nβ) has previously been proved for diffeomorphisms, we establish the decay rate...
متن کاملEuclidean Formulation of Discrete Uniformization of the Disk
Thurston’s circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to the circle. The main proofs of the uniformization use hyperbolic volumes (Andr...
متن کاملDynamical Coherence of Partially Hyperbolic Diffeomorphisms of Tori Isotopic to Anosov
We show that partially hyperbolic diffeomorphisms of d-dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anoso...
متن کاملGeodesic Flows on Diffeomorphisms of the Circle, Grassmannians, and the Geometry of the Periodic KdV Equation
We start by constructing a Hilbert manifold T of orientation preserving diffeomorphisms of the circle (modulo the group of bi-holomorphic self-mappings of the disc). This space, which could be thought of as a completion of the universal Teichmüller space, is endowed with a right-invariant Kähler metric. Using results from the theory of quasiconformal mappings we construct an embedding of T into...
متن کامل