Semiconjugacies between the Julia sets of geometrically finite rational maps

Kawahira, Tomoki

doi:10.1017/s0143385702001682

Cited by 9 publications

(6 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Furthermore, the conjugacy depends continuously on g and tends to the identity as g tends to f along γ. See [CT18], [Ha98,Ha00,Ha02] and [Kaw03,Kaw06] for more details.…”

Section: Theoremmentioning

confidence: 99%

Geometrically finite transcendental entire functions

Alhamed,

Rempe,

Sixsmith

2020

Preprint

View full text Add to dashboard Cite

For polynomials, local connectivity of Julia sets is an important and muchstudied property, as it leads to a complete description of the topological dynamics as a quotient of a much simpler system, namely angle d-tupling on the circle, where d ≥ 2 is the degree.For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To do so, we introduce the notion of docile functions: a transcendental entire function with bounded postsingular set is docile if its dynamics can be described as the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This can be seen as an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author, and of Mihaljević-Brandt, for more restrictive classes of entire functions.Our techniques allow us deduce a number of other results for geometrically finite functions with bounded criticality, concerning bounded Fatou components and the local connectivity of Julia sets. In particular, we show that the Julia set of the sine function is locally connected, answering a question raised by Osborne.

show abstract

“…Furthermore, the conjugacy depends continuously on g and tends to the identity as g tends to f along γ. See [CT18], [Ha98,Ha00,Ha02] and [Kaw03,Kaw06] for more details.…”

Section: Theoremmentioning

confidence: 99%

Geometrically finite transcendental entire functions

Alhamed,

Rempe,

Sixsmith

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…, β l ′ } has multiplier ω ′ = exp(2πp ′ /q ′ ). By applying Theorem 1.1 of [Ka1] to (f → g), we have:…”

Section: A Appendixmentioning

confidence: 99%

“…(See also Proposition 2.1.) The proof of Theorem 1.1 of [Ka1] is based on a pull-back argument and it does not use quasiconformal maps. Here is a useful corollary which easily follows from property 3:…”

Section: A Appendixmentioning

confidence: 99%

Tessellation and Lyubich–Minsky laminations associated with quadratic maps, I: pinching semiconjugacies

Kawahira¹

2009

Ergod. Th. Dynam. Sys.

Self Cite

View full text Add to dashboard Cite

We introduce tessellation of the filled Julia sets for hyperbolic and parabolic quadratic maps. Then the dynamics inside their Julia sets are organized by tiles which work like external rays outside. We also construct continuous families of pinching semiconjugacies associated with hyperblic-to-parabolic degenerations without using quasiconformal deformation. Instead we use tessellation and investigation on the hyperbolic-to-parabolic degeneration of linearizing coordinates inside the Julia set.

show abstract

“…Furthermore, the conjugacy depends continuously on

g

and tends to the identity as

g

tends to

f

along

\gamma

. See [11, 15–17, 21, 22] for more details.…”

Section: Introductionmentioning

confidence: 99%

Geometrically finite transcendental entire functions

Alhamed

Rempe

Sixsmith

2022

Journal of London Math Soc

View full text Add to dashboard Cite

For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree 𝑑 ⩾ 2 is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle 𝑑-tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of docile functions (Definition 1.1): a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This is an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljević for more restrictive classes of entire functions. We deduce a number of further results, concerning bounded Fatou components and the local connectivity of Julia sets. In particular, we show that the Julia set of the sine function is locally connected, answering a question raised by Osborne.

show abstract

Semiconjugacies between the Julia sets of geometrically finite rational maps

Cited by 9 publications

References 10 publications

Geometrically finite transcendental entire functions

Geometrically finite transcendental entire functions

Tessellation and Lyubich–Minsky laminations associated with quadratic maps, I: pinching semiconjugacies

Geometrically finite transcendental entire functions

Contact Info

Product

Resources

About