American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021144
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Criniferous entire maps with absorbing Cantor bouquets

 Department of Mathematics, The University of Manchester, Manchester, M13 9PL, UK

Received  October 2020 Revised  July 2021 Early access October 2021

It is known that, for many transcendental entire functions in the Eremenko-Lyubich class $\mathcal{B}$, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are criniferous. In this paper, we extend this result to a new class of maps in $\mathcal{B}$. Furthermore, we show that if a map belongs to this class, then its Julia set contains a Cantor bouquet; in other words, it is a subset of $\mathbb{C}$ ambiently homeomorphic to a straight brush.

Citation: Leticia Pardo-Simón. Criniferous entire maps with absorbing Cantor bouquets. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021144
References:

show all references

References:
On the left, hairs of a Cantor bouquet intersecting a circle $\partial {{\mathbb{D}}}_R$, some of them multiple times. For each hair, dashes represent points with lower potential than that of the last point that intersects $\partial {{\mathbb{D}}}_R$. On the right, the image of the hairs to a straight brush under an ambient homeomorphism $\psi$. $[-Q, Q]^2$ is a square whose boundary the hairs intersect at most once, and $S_R: = \psi^{-1}((-Q, Q)^2)$
Construction of a neighbourhood of $z_n(\eta)$ in Claim Claim 2 by pulling back balls centred at $f^j(z_n)$ for all $1\leq j\leq n$ such that $f^j(z_n) \in \partial S_R$
Proof of Proposition 8 by interpolating the maps $\psi_g$ and ${\varphi}_f$ using the annulus $\mathcal{R}$ shown in orange
 [1] Núria Fagella, David Martí-Pete. Dynamic rays of bounded-type transcendental self-maps of the punctured plane. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3123-3160. doi: 10.3934/dcds.2017134 [2] Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321 [3] Patricia Domínguez, Peter Makienko, Guillermo Sienra. Ruelle operator and transcendental entire maps. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 773-789. doi: 10.3934/dcds.2005.12.773 [4] Koh Katagata. Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5319-5337. doi: 10.3934/dcds.2019217 [5] Piermarco Cannarsa, Peter R. Wolenski. Semiconcavity of the value function for a class of differential inclusions. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 453-466. doi: 10.3934/dcds.2011.29.453 [6] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3337-3349. doi: 10.3934/dcdss.2020443 [7] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3821-3836. doi: 10.3934/dcdss.2020436 [8] S. Astels. Thickness measures for Cantor sets. Electronic Research Announcements, 1999, 5: 108-111. [9] Igor Shevchenko, Barbara Kaltenbacher. Absorbing boundary conditions for the Westervelt equation. Conference Publications, 2015, 2015 (special) : 1000-1008. doi: 10.3934/proc.2015.1000 [10] Jianxun Fu, Song Zhang. A new type of non-landing exponential rays. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4179-4196. doi: 10.3934/dcds.2020177 [11] David Cheban. Belitskii--Lyubich conjecture for $C$-analytic dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 945-959. doi: 10.3934/dcdsb.2015.20.945 [12] Jihoon Lee, Vu Manh Toi. Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3135-3152. doi: 10.3934/dcdsb.2020054 [13] Mehdi Pourbarat. On the arithmetic difference of middle Cantor sets. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4259-4278. doi: 10.3934/dcds.2018186 [14] Maik Gröger, Olga Lukina. Measures and stabilizers of group Cantor actions. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2001-2029. doi: 10.3934/dcds.2020350 [15] Xilin Fu, Zhang Chen. New discrete analogue of neural networks with nonlinear amplification function and its periodic dynamic analysis. Conference Publications, 2007, 2007 (Special) : 391-398. doi: 10.3934/proc.2007.2007.391 [16] Hongming Yang, C. Y. Chung, Xiaojiao Tong, Pingping Bing. Research on dynamic equilibrium of power market with complex network constraints based on nonlinear complementarity function. Journal of Industrial & Management Optimization, 2008, 4 (3) : 617-630. doi: 10.3934/jimo.2008.4.617 [17] Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021 [18] Agnieszka Badeńska. Measure rigidity for some transcendental meromorphic functions. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2375-2402. doi: 10.3934/dcds.2012.32.2375 [19] E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401 [20] Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251

2020 Impact Factor: 1.392

Tools

Article outline

Figures and Tables