Parabolic Cohomology of Arithmetic Subgroups of SL(2, Z ) with Coefficients in the Field of Rational Functions on the Riemann Sphere

1989 ◽  
Vol 111 (1) ◽  
pp. 35 ◽  
Author(s):  
Avner Ash
Keyword(s):  
Riemann Sphere ◽  
Rational Functions

Possibility of Uniform Rational Approximation in the Spherical Metric

1976 ◽  
Vol 28 (1) ◽  
pp. 112-115 ◽  
Author(s):  
P. M. Gauthier ◽  
A. Roth ◽  
J. L. Walsh
Keyword(s):  
Compact Subset ◽  
Rational Approximation ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Finite Complex ◽  
Finite Plane ◽  
Extended Plane ◽  
Chordal Metric

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.

Complex dynamics of Möbius semigroups

2012 ◽  
Vol 32 (6) ◽  
pp. 1889-1929 ◽  
Author(s):  
DAVID FRIED ◽  
SEBASTIAN M. MAROTTA ◽  
RICH STANKEWITZ
Keyword(s):  
Complex Dynamics ◽  
Riemann Sphere ◽  
Julia Sets ◽  
Fibonacci Sequence ◽  
Rational Functions ◽  
Möbius Transformations ◽  
Random Dynamics ◽  
Compare And Contrast ◽  
Möbius Groups ◽  
Mobius Transformations

AbstractWe study the dynamics of semigroups of Möbius transformations on the Riemann sphere, especially their Julia sets and attractors. This theory relates to the dynamics of rational functions, rational semigroups, and Möbius groups and we compare and contrast these theories. We particularly examine Caruso’s family of Möbius semigroups, based on a random dynamics variant of the Fibonacci sequence.

On the transfer operator for rational functions on the Riemann sphere

1996 ◽  
Vol 16 (2) ◽  
pp. 255-266 ◽  
Author(s):  
Manfred Denker ◽  
Feliks Przytycki ◽  
Mariusz Urbański
Keyword(s):  
Equilibrium Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Transfer Operator ◽  
Rational Functions ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Correlation Integral ◽  
Hölder Continuous Functions ◽  
Conformal Measure

AbstractLet T be a rational function of degree ≥ 2 on the Riemann sphere. Our results are based on the lemma that the diameter of a connected component of T−n(B(x, r)), centered at any point x in its Julia set J = J(T), does not exceed Lnrp for some constants L ≥ 1 and ρ > 0. Denote the transfer operator of a Hölder-continuous function φ on J satisfying P(T,φ) > supz∈Jφ(z). We study the behavior of {: n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) normbounded in the space of Hölder-continuous functions for sufficiently small exponent. As a consequence we obtain that the density of the equilibrium measure μ for φ with respect to the -conformal measure is Höolder-continuous. We also prove that the rate of convergence of {ψ to this density in sup-norm is . From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.

scholarly journals Recurrence rates for loosely Markov dynamical systems

2007 ◽  
Vol 82 (1) ◽  
pp. 39-57 ◽  
Author(s):  
Mariusz Urbański
Keyword(s):  
Dynamical Systems ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Iterated Function Systems ◽  
Recurrence Rates ◽  
Exponential Maps ◽  
Iterated Function

AbstractThe concept of loosely Markov dynamical systems is introduced. We show that for these systems the recurrence rates and pointwise dimensions coincide. The systems generated by hyperbolic exponential maps, arbitrary rational functions of the Riemann sphere, and measurable dynamical systems generated by infinite conformal iterated function systems are all checked to be loosely Markov.

Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products

2001 ◽  
Vol 21 (2) ◽  
pp. 563-603 ◽  
Author(s):  
HIROKI SUMI
Keyword(s):  
Hausdorff Dimension ◽  
Riemann Sphere ◽  
Julia Set ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Empty Interior ◽  
Skew Products ◽  
Wandering Domain ◽  
Upper Estimate ◽  
Interior Points

We consider dynamics of sub-hyperbolic and semi-hyperbolic semigroups of rational functions on the Riemann sphere and will show some no wandering domain theorems. The Julia set of a rational semigroup in general may have non-empty interior points. We give a sufficient condition that the Julia set has no interior points. From some information about forward and backward dynamics of the semigroup, we consider when the area of the Julia set is equal to zero or an upper estimate of the Hausdorff dimension of the Julia set.

Unwinding paths on the Riemann sphere for continuous integrals of rational functions

2015 ◽  
Vol 49 (1) ◽  
pp. 35-35 ◽  
Author(s):  
Robert H. C. Moir ◽  
Robert M. Corless ◽  
David J. Jeffrey
Keyword(s):  
Riemann Sphere ◽  
Rational Functions

scholarly journals On Rational Functions with More than Three Branch Points

2020 ◽  
Vol 27 (02) ◽  
pp. 231-246
Author(s):  
Jijian Song ◽  
Bin Xu
Keyword(s):  
Positive Integer ◽  
Rational Function ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Branch Points ◽  
New Class ◽  
Belyi Functions

Let d be a positive integer and Λ be a collection of partitions of d of the form (a1, …, ap), (b1, …, bq), (m1 + 1, 1, …, 1), …, (ml + 1, 1, …, 1), where (m1, …, ml) is a partition of p + q − 2 > 0. We prove that there exists a rational function on the Riemann sphere with branch data Λ if and only if max(m1, …, ml) < d/GCD(a1, …, ap, b1, …, bq). As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.

Rational functions with no recurrent critical points

1994 ◽  
Vol 14 (2) ◽  
pp. 391-414 ◽  
Author(s):  
Mariusz Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Critical Points ◽  
Riemann Sphere ◽  
Julia Set ◽  
Sufficient Conditions ◽  
Rational Functions ◽  
Packing Measure ◽  
Rational Map ◽  
Dimensional Hausdorff Measure ◽  
Necessary And Sufficient

AbstractLet h be the Hausdorff dimension of the Julia set of a rational map with no nonperiodic recurrent critical points. We give necessary and sufficient conditions for h-dimensional Hausdorff measure and h-dimensional packing measure of the Julia set to be positive and finite. We also show that either the Julia set is the whole Riemann sphere or h < 2 and that if a rational map (not necessarily with no nonperiodic recurrent critical points!) has a rationally indifferent periodic point, then h > 1/2.

scholarly journals Fine inducing and equilibrium measures for rational functions of the Riemann sphere

2015 ◽  
Vol 210 (1) ◽  
pp. 399-465 ◽  
Author(s):  
Michał Szostakiewicz ◽  
Mariusz Urbański ◽  
Anna Zdunik
Keyword(s):  
Riemann Sphere ◽  
Rational Functions ◽  
Equilibrium Measures
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