Rational maps and Kleinian groups
نویسنده
چکیده
What are the possible topological forms for a conformal dynamical system? Part of the answer is provided by two theorems, due to Thurston, which employ iteration on Teichmüller space to construct rational maps and Kleinian groups of a given topological form. More precisely, the iteration either finds a geometric model or reveals a topological obstruction to its existence. This dichotomy stems from:
منابع مشابه
Some Rational Maps Whose Julia Sets Are Not Locally Connected
We describe examples of rational maps which are not topologically conjugate to a polynomial and whose Julia sets are connected but not locally connected. Introduction and motivations The dynamics of a rational map f acting on Ĉ is concentrated on its Julia set which is (by definition) the minimal compact set invariant by f and f−1 containing at least three points. The question of local connecti...
متن کاملHausdorff dimension and conformal dynamics III: Computation of dimension
This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Dimension graphs are presented for (a) the family of Fuchsian groups generated by reflections in 3 ...
متن کاملCannon–thurston Maps for Kleinian Groups
We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show...
متن کاملHausdorr Dimension and Conformal Dynamics Ii: Geometrically Nite Rational Maps
This paper investigates several dynamically de ned dimensions for rational maps f on the Riemann sphere, and gives a systematic development modeled on the theory for Kleinian groups. The radial Julia set is de ned and we show H: dim(Jrad(f)) = (f), the minimal dimension of an f -invariant density. The map f is geometrically nite if every critical point in the Julia set is preperiodic. In this c...
متن کاملGroup Algebras of Kleinian Type and Groups of Units
The algebras of Kleinian type are finite dimensional semisimple rational algebras A such that the group of units of an order in A is commensurable with a direct product of Kleinian groups. We classify the Schur algebras of Kleinian type and the group algebras of Kleinian type. As an application, we characterize the group rings RG, with R an order in a number field and G a finite group, such tha...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1990