Semigroups of Isometries of the Hyperbolic Plane

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.

Parameter spaces of locally constant cocycles

“…We note that the elliptic locus was originally defined in [24] and [1] as the set of N -tuples whose semigroup contains an elliptic matrix (and not the identity), and Qustion 1 was stated with this definition in mind. However, the answer to the original statement of Question 1 was shown to be negative by Jacques and Short [17,Section 16]. Question 1 as we have stated it is a restatement of the questions of Yoccoz and Avila, Bochi and Yoccoz by Jacques and Short (see the last part of [17,Section 16]).…”

“…However, the answer to the original statement of Question 1 was shown to be negative by Jacques and Short [17,Section 16]. Question 1 as we have stated it is a restatement of the questions of Yoccoz and Avila, Bochi and Yoccoz by Jacques and Short (see the last part of [17,Section 16]). We also note that not all results from [24] and [1] hold for the elliptic locus as defined in Definition 1.2; for example our elliptic locus is no longer an open set.…”

“…Semigroups of Möbius transformations have been previously studied by Fried, Marotta and Stankiewitz [14], as a particular branch of the theory of semigroups of rational functions initiated by Hinkkanen and Martin [16]. Our analysis is closely related to the work of Jacques and Short [17] who further developed the material in [14] by incorporating well-known results from the theory of Fuchsian groups.…”

“…Comparing Theorems 2.1 and 3.3 we can see a geometric interpretation of the fact that the hyperbolic locus H is contained in the semidiscrete and inverse-free locus S. The two loci however are certainly not identical as the N -tuples in the following lemma, taken from [17,Lemma 10.4], demonstrate.…”

“…The semidiscrete and inverse-free properties were introduced by Jacques and Short in [17], where they were used in order to study semigroups of isometries of the hyperbolic plane. Observe that an N -tuple A is semidiscrete and inverse-free if and only if I / ∈ A where I is the 2 × 2 identity matrix.…”