Conical limit sets and continued fractions

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.

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“…It is straightforward to prove this theorem using hyperbolic geometry; see, for example, [11,Proposition 3.3].…”

Section: Limit Setsmentioning

confidence: 99%

Semigroups of Isometries of the Hyperbolic Plane

Jacques

Short

2020

International Mathematics Research Notices

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“…It is known that if the backward limit set of a restrained sequence of Möbius transformations contains an open disc E, then the conical limit set is uncountable and its closure contains E (see [6,Lem. 5.12]).…”

Section: 32])mentioning

confidence: 99%

The Parabola Theorem on Continued Fractions

Short

2016

Comput. Methods Funct. Theory

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Using geometric methods borrowed from the theory of Kleinian groups, we interpret the parabola theorem on continued fractions in terms of sequences of Möbius transformations. This geometric approach allows us to relate the Stern-Stolz series, which features in the parabola theorem, to the dynamics of certain sequences of Möbius transformations acting on three-dimensional hyperbolic space. We also obtain a version of the parabola theorem in several dimensions.

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“…The set of divergence of S n is the set of points z in C ∞ for which the sequence S n (z) diverges. Sets of divergence for general sequences of Möbius transformations have been studied in [5,6,14]. The set of divergence is closely related to the limit set from Kleinian group theory and the Julia set from complex dynamics.…”

Section: Introductionmentioning

confidence: 99%

Hausdorff dimension of sets of divergence arising from continued fractions

Short

2012

Proc. Amer. Math. Soc.

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Abstract. A complex continued fraction can be represented by a sequence of Möbius transformations in such a way that the continued fraction converges if and only if the sequence converges at the origin. The set of divergence of the sequence of Möbius transformations is equivalent to the conical limit set from Kleinian group theory, and it is closely related to the Julia set from complex dynamics. We determine the Hausdorff dimensions of sets of divergence for sequences of Möbius transformations corresponding to certain important classes of continued fractions.

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Conical limit sets and continued fractions

Cited by 3 publications

References 14 publications

Semigroups of Isometries of the Hyperbolic Plane

Semigroups of Isometries of the Hyperbolic Plane

The Parabola Theorem on Continued Fractions

Hausdorff dimension of sets of divergence arising from continued fractions

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