Complex dynamics of Möbius semigroups

Fried, David; Marotta, Sebastían M.; Stankewitz, Rich

doi:10.1017/s014338571100054x

Cited by 6 publications

(25 citation statements)

References 30 publications

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“…(2) is shown in [11]. Because E(G) is clearly backward invariant, (3) follows from Proposition 2.17 in [8] when E(G) is finite. Now suppose E(G) is infinite.…”

Section: Propositionmentioning

confidence: 89%

“…We define the Möbius semigroup S β = f, g , and its inverse semigroup S ′ β = f −1 , g −1 . See [8] for background, but note that there the notation is different in that Möbius groups are denoted by . .…”

Section: Applicationsmentioning

confidence: 99%

“…and semigroups are denoted ≺ · · · ≻. We note that in Section 7 of [8] it is shown that neither S β nor S ′ β have finite orbits in H 3 ∪ C and so, in particular, E(S ′ β ) = ∅ and E(S β ) = ∅. Further it is shown that #J(S β ) ≥ 3 and #J(S ′ β ) ≥ 3.…”

Section: Applicationsmentioning

confidence: 99%

“…However, they studied only rational semigroups containing at least one element of degree at least two. For an in depth look at the dynamics of Möbius semigroups see [8].…”

Section: Introductionmentioning

confidence: 99%

“…

We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick attractor in the sense given in [8].

…”

mentioning

confidence: 96%

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Backward iteration algorithms for Julia sets of Möbius semigroups

Stankewitz¹,

Sumi²

2016

DCDS-A

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“…(2) is shown in [11]. Because E(G) is clearly backward invariant, (3) follows from Proposition 2.17 in [8] when E(G) is finite. Now suppose E(G) is infinite.…”

Section: Propositionmentioning

confidence: 89%

Section: Applicationsmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

“…However, they studied only rational semigroups containing at least one element of degree at least two. For an in depth look at the dynamics of Möbius semigroups see [8].…”

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 96%

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Backward iteration algorithms for Julia sets of Möbius semigroups

Stankewitz¹,

Sumi²

2016

DCDS-A

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Semigroups of Isometries of the Hyperbolic Plane

Jacques

Short

2020

International Mathematics Research Notices

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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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