Construction of Some Curious Diffeomorphisms of the Riemann Sphere
نویسندگان
چکیده
Notation We denote the Riemann sphere C U {00} by § 2 and for a e U, we denote by R a :z >e 2nia z the rotation of § 2 of rotation number a with fixed points 0 and 00. On S 2 c W we put the standard induced metric d; and for / > 0 the /-ball with centre x is {ye § 2 :d(x, y) < /}. On subsets of § 2 we put the induced topology. Given A c= § 2 we denote the e-neighbourhood of A by = {ye §*:d(y,A)<e}. We denote the unit disk by B = {zeC:|z|< 1}. If we write p/qeQ we shall suppose that q ^ 1, and p and q are relatively prime.
منابع مشابه
Spatial statistics for lattice points on the sphere I: Individual results
We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares. We examine several statistics of these point sets, such as the electrostatic potential, Ripley's function, the variance of the number of points in random spherical caps, and the covering radius. Some of the results are conditional on t...
متن کاملTopological Pressure for One-dimensional Holomorphic Dynamical Systems
For a class of one-dimensional holomorphic maps f of the Riemann sphere we prove that for a wide class of potentials φ the topological pressure is entirely determined by the values of φ on the repelling periodic points of f . This is a version of a classical result of Bowen for hyperbolic diffeomorphisms in the holomorphic non-uniformly hyperbolic setting.
متن کاملRenormalization in quantum field theory and the Riemann - Hilbert problem II : the β - function ,
We showed in part I that the Hopf algebra H of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ε = 0 the holomorphic part γ+(ε) of the Riemann–Hilbert decomposition γ−(ε) γ+(ε) of the loop γ(ε) ∈ G provided by dimensional regularization. We show in ...
متن کاملIntegrating the Chirally Split Diffeomorphism Anomaly on a Compact Riemann Surface †
A well-defined chirally split functional integrating the 2D chirally split diffeomorphism anomaly is exhibited on an arbitrary compact Riemann surface without boundary. The construction requires both the use of the Beltrami parametrisation of complex structures and the introduction of a background metric possibly subject to a Liouville equation. This formula reproduces in the flat case the so-c...
متن کاملThe construction of analytic diffeomorphisms for exact robot navigation on star worlds
A navigation function is a scalar valued function on a robot configuration space which encodes the task of moving to a desired destination without hitting any obstacles. Our program of research concerns the construction of navigation functions on a family of configuration spaces whose “geometric expressiveness” is rich enough for navigation amidst real world obstacles. A sphere world is a compa...
متن کامل