Random iteration of Möbius transformations and Furstenberg's theorem

2000 ◽  
Vol 20 (4) ◽  
pp. 953-962 ◽  
Author(s):  
AMIRAN AMBROLADZE ◽  
HANS WALLIN
Keyword(s):  
Probability Measure ◽  
Half Plane ◽  
Initial Point ◽  
Möbius Transformations ◽  
Random Maps ◽  
Random Iteration ◽  
Upper Half Plane ◽  
Set Up ◽  
Mobius Transformations

Let $Y_1, Y_2, \dots$ be a sequence of independent random maps, identically distributed with respect to a probability measure $\mu$ on $SL(2,R)$. A (deep) theorem of Furstenberg gives abstract conditions under which for almost every such sequence the orbit of a non-zero initial point in $R^2$ tends to infinity exponentially fast. In the present paper we translate this statement into the set-up of Möbius transformations on the upper half-plane and provide a very explicit way to determine whether or not the required conditions are satisfied.

scholarly journals Semigroups of Isometries of the Hyperbolic Plane

2020 ◽  
Author(s):  
Matthew Jacques ◽  
Ian Short
Keyword(s):  
Accumulation Point ◽  
Finite Collection ◽  
Finitely Generated ◽  
Limit Sets ◽  
Möbius Transformations ◽  
Möbius Group ◽  
Upper Half Plane ◽  
Open Question ◽  
The Relationship ◽  
Mobius Transformations

Abstract Motivated by a problem on the dynamics of compositions of plane hyperbolic isometries, we prove several fundamental results on semigroups of isometries, thought of as real Möbius transformations. We define a semigroup $S$ of Möbius transformations to be semidiscrete if the identity map is not an accumulation point of $S$. We say that $S$ is inverse free if it does not contain the identity element. One of our main results states that if $S$ is a semigroup generated by some finite collection $\mathcal{F}$ of Möbius transformations, then $S$ is semidiscrete and inverse free if and only if every sequence of the form $F_n=f_1\dotsb f_n$, where $f_n\in \mathcal{F}$, converges pointwise on the upper half-plane to a point on the ideal boundary, where convergence is with respect to the chordal metric on the extended complex plane. We fully classify all two-generator semidiscrete semigroups and include a version of Jørgensen’s inequality for semigroups. We also prove theorems that have familiar counterparts in the theory of Fuchsian groups. For instance, we prove that every semigroup is one of four standard types: elementary, semidiscrete, dense in the Möbius group, or composed of transformations that fix some nontrivial subinterval of the extended real line. As a consequence of this theorem, we prove that, with certain minor exceptions, a finitely generated semigroup $S$ is semidiscrete if and only if every two-generator semigroup contained in $S$ is semidiscrete. After this we examine the relationship between the size of the “group part” of a semigroup and the intersection of its forward and backward limit sets. In particular, we prove that if $S$ is a finitely generated nonelementary semigroup, then $S$ is a group if and only if its two limit sets are equal. We finish by applying some of our methods to address an open question of Yoccoz.

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

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).

Exact determination of gravity stresses in finite elastic slopes: Part I. Theoretical considerations

1983 ◽  
Vol 20 (1) ◽  
pp. 47-54 ◽  
Author(s):  
V. Silvestri ◽  
C. Tabib
Keyword(s):  
Plane Strain ◽  
Half Plane ◽  
Inclination Angle ◽  
Complex Variable ◽  
Earth Pressure ◽  
Exact Distributions ◽  
Exact Determination ◽  
Upper Half Plane ◽  
Theoretical Considerations

The exact distributions of gravity stresses are obtained within slopes of finite height inclined at various angles, −β (β = π/2, π/3, π/4, π/6, and π/8), to the horizontal. The solutions are obtained by application of the theory of a complex variable. In homogeneous, isotropic, and linearly elastic slopes under plane strain conditions, the gravity stresses are independent of Young's modulus and are a function of (a) the coordinates, (b) the height, (c) the inclination angle, (d) Poisson's ratio or the coefficient of earth pressure at rest, and (e) the volumetric weight. Conformal applications that transform the planes of the various slopes studied onto the upper half-plane are analytically obtained. These solutions are also represented graphically.

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