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.

scholarly journals Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles

2010 ◽  
Vol 30 (6) ◽  
pp. 1869-1902 ◽  
Author(s):  
HIROKI SUMI
Keyword(s):  
Jordan Curve ◽  
Riemann Sphere ◽  
Julia Set ◽  
Julia Sets ◽  
John Domain ◽  
Unbounded Component ◽  
Random Dynamics ◽  
Jordan Curves ◽  
Polynomial Maps

AbstractWe investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere $\CCI $) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups G such that G is generated by a compact family Γ, the planar postcritical set of G is bounded, and G is (semi-) hyperbolic. In one of the classes, we have that, for almost every sequence $\gamma \in \Gamma ^{\NN }$, the Julia set Jγ of γ is a Jordan curve but not a quasicircle, the unbounded component of $\CCI {\setminus } J_{\gamma }$ is a John domain, and the bounded component of $\CC {\setminus } J_{\gamma }$ is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups G such that the planar postcritical set of G is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.

scholarly journals Visualization of Mandelbrot and Julia Sets of Möbius Transformations

2021 ◽  
Vol 5 (3) ◽  
pp. 73
Author(s):  
Leah K. Mork ◽  
Darin J. Ulness
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Iteration Step ◽  
Möbius Transformation ◽  
Möbius Transformations ◽  
Fractal Sets ◽  
Mandelbrot Sets ◽  
Möbius Transforms ◽  
Mobius Transformations

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

scholarly journals Möbius geometry for hypersurfaces in S4

1995 ◽  
Vol 139 ◽  
pp. 1-20 ◽  
Author(s):  
Changping Wang
Keyword(s):  
Invariant System ◽  
Principal Curvatures ◽  
Möbius Transformations ◽  
Möbius Geometry ◽  
Mobius Transformations

Our purpose in this paper is to study Möbius geometry for those hypersurfaces in S4 which have different principal curvatures at each point. We will give a complete local Möbius invariant system for such hypersurface in S4 which determines the hypersurface up to Möbius transformations. And we will classify the so-called Möbius homogeneous hypersurfaces in S4.

MÖBIUS TRANSFORMATIONS IN QUANTUM AMPLITUDE AMPLIFICATION WITH GENERALIZED PHASES

2010 ◽  
Vol 08 (06) ◽  
pp. 923-935 ◽  
Author(s):  
CÉSAR BAUTISTA-RAMOS ◽  
NORA CASTILLO-TÉPOX
Keyword(s):  
Error Probability ◽  
Quantum Algorithm ◽  
Linear Transformations ◽  
Möbius Transformations ◽  
Good State ◽  
Quantum Amplitude ◽  
Initial States ◽  
Phase Angles ◽  
Amplitude Amplification ◽  
Mobius Transformations

The iteration of the operators employed in quantum amplitude amplification with generalized phases is analyzed by using elementary properties (geometric and algebraic) of the Möbius transformations (fractional linear transformations). It is shown that, for a given quantum algorithm without measurement, which produces a good state with probability a of success, if the phase angles φ and ϕ which mark the good and initial states respectively satisfy φ = ϕ with a small enough, then, for a number n of iterations with [Formula: see text] we get an error probability that is at most O(aϕ2).

scholarly journals Ellipses, near ellipses, and harmonic Möbius transformations

2005 ◽  
Vol 133 (9) ◽  
pp. 2705-2710 ◽  
Author(s):  
Martin Chuaqui ◽  
Peter Duren ◽  
Brad Osgood
Keyword(s):  
Möbius Transformations ◽  
Mobius Transformations
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