Topology and measure of buried points in Julia sets

Curry, Clinton P.; Mayer, John C.; Tymchatyn, E. D.

doi:10.4064/fm222-1-1

Cited by 4 publications

(2 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following result of [3] is a significant consequence of Theorem 1. As noted in [3], it follows from Theorem 1.3 that…”

Section: Introductionmentioning

confidence: 79%

The maximal entropy measure of Fatou boundaries

Hawkins¹,

Taylor²

2018

Discrete &Amp; Continuous Dynamical Systems - A

View full text Add to dashboard Cite

We look at the maximal entropy (MME) measure of the boundaries of connected components of the Fatou set of a rational map of degree ≥ 2. We show that if there are infinitely many Fatou components, and if either the Julia set is disconnected or the map is hyperbolic, then there can be at most one Fatou component whose boundary has positive MME measure. We also replace hyperbolicity by the more general hypothesis of geometric finiteness.

show abstract

“…The following result of [3] is a significant consequence of Theorem 1. As noted in [3], it follows from Theorem 1.3 that…”

Section: Introductionmentioning

confidence: 79%

The maximal entropy measure of Fatou boundaries

Hawkins¹,

Taylor²

2018

Discrete &Amp; Continuous Dynamical Systems - A

View full text Add to dashboard Cite

show abstract

“…Makienko conjectured that the Julia set of a rational map f has buried points if and only if there is no completely invariant component of the Fatou set of f •2 (see [Mak87]). One can refer to [Mor97], [Qia97], [Mor00], [SY03], [CM 2 R09], [CMT13] and the references therein for the progress. 1 The quadratic rational map cannot contain any buried Julia component since its Julia set is either connected or totally disconnected.…”

Section: Introductionmentioning

confidence: 99%

Julia sets as buried Julia components

Wang

Yang

2020

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Let f be a rational map with degree d ≥ 2 whose Julia set is connected but not equal to the whole Riemann sphere. It is proved that there exists a rational map g such that g contains a buried Julia component on which the dynamics is quasiconformally conjugate to that of f on the Julia set if and only if f does not have parabolic basins and Siegel disks. If such g exists, then the degree can be chosen such that deg(g) ≤ 7d − 2. In particular, if f is a polynomial, then g can be chosen such that deg(g) ≤ 4d + 4. Moreover, some quartic and cubic rational maps whose Julia sets contain buried Jordan curves are also constructed. 3. From semi-buried to buried 13 3.1. Cutting along the equipotentials II 13 3.2. Annulus-to-disk and annulus-to-annulus coverings II 16 3.3. Uniformization II 16 3.4. The buried property 17 4. The first case of exactly one attracting cycle 18 5. The second case of exactly one attracting cycle: dendrite 19 6. The quartic Julia sets containing buried Jordan curves 23 6.1. The position of the critical orbits 23 6.2. The Julia set moves holomorphically 24 6.3. The semi-buried component and buried Jordan curves 26 7. The cubic Julia sets containing buried Jordan curves 27 References

show abstract

Maximal Entropy Measures on Julia Sets and a Computer Algorithm

Hawkins

2021

Ergodic Dynamics

View full text Add to dashboard Cite

Topology and measure of buried points in Julia sets

Cited by 4 publications

References 22 publications

The maximal entropy measure of Fatou boundaries

The maximal entropy measure of Fatou boundaries

Julia sets as buried Julia components

Maximal Entropy Measures on Julia Sets and a Computer Algorithm

Contact Info

Product

Resources

About