On the pullback relation on curves induced by a Thurston map

Pilgrim, Kevin M.

doi:10.48550/arxiv.2102.12402

Cited by 2 publications

(2 citation statements)

References 7 publications

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“…In particular, we are able to describe the iterative behavior of µ f for a specific obstructed Thurston map f obtained by blowing up the (2 × 2)-Lattès map (see Section 9.2 for the details). This provides an answer to a question by Kevin Pilgrim (see [Pil21,Question 4.4]).…”

Section: Introductionmentioning

confidence: 89%

Eliminating Thurston obstructions and controlling dynamics on curves

Bonk,

Hlushchanka,

Iseli

2021

Preprint

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Every Thurston map f ∶ S 2 → S 2 on a 2-sphere S 2 induces a pull-back operation on Jordan curves α ⊂ S 2 ∖ P f , where P f is the postcritical set of f . Here the isotopy class [f −1 (α)] (relative to P f ) only depends on the isotopy class [α]. We study this operation for Thurston maps with four postcritical points. In this case a Thurston obstruction for the map f can be seen as a fixed point of the pull-back operation.We show that if a Thurston map f with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can "blow up" suitable arcs in the underlying 2-sphere and construct a new Thurston map f for which this obstruction is eliminated. We prove that no other obstruction arises and so f is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points.We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.

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Section: Introductionmentioning

confidence: 89%

Eliminating Thurston obstructions and controlling dynamics on curves

Bonk,

Hlushchanka,

Iseli

2021

Preprint

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“…In §4.1, we shall discuss this example and some possible generalizations of it. It is an open question if the pullback can be constant for a Thurston map of a prime degree [Pil21].…”

Section: The Case Of Constant Pullbackmentioning

confidence: 99%

On the rank of the Thurston pullback map

Filom¹

2022

Preprint

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Under some mild assumptions, an orientation-preserving branched covering map of marked 2-spheres induces a pullback map between the corresponding Teichmüller spaces. By analyzing the associated pushforward operator acting on integrable quadratic differentials, we obtain a global lower bound on the rank of the derivative of the pullback map in terms of the action of the cover on marked points. In the dynamical context, the two sets of marked points in the target and source coincide with the postcritical set. Investigating the resulting pullback map is the central part of Thurston's topological characterization of postcritically finite rational maps. Postcritically finite maps with constant pullback have been studied by various authors. In that direction, our approach provides upper bounds on the size of the postcritical set of a map with constant pullback, and shows that the postcritical dynamics is highly restricted.

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On the pullback relation on curves induced by a Thurston map

Cited by 2 publications

References 7 publications

Eliminating Thurston obstructions and controlling dynamics on curves

Eliminating Thurston obstructions and controlling dynamics on curves

On the rank of the Thurston pullback map

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