American Institute of Mathematical Sciences

Advanced
April  2005, 13(1): 13-34. doi: 10.3934/dcds.2005.13.13

Necessary conditions for the existence of wandering triangles for cubic laminations

Alexander Blokh 1,

1. 

Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, AL 35294-1170, United States

Received  March 2004 Revised  November 2004 Published  March 2005

In his 84 preprint W. Thurston proved that quadratic laminations do not admit so-called wandering triangles and asked a deep question concerning their existence for laminations of higher degrees. Recently it has been discovered by L. Oversteegen and the author that some closed laminations of the unit circle invariant under $z\mapsto z^d, d>2$ admit wandering triangles. This makes the problem of describing the criteria for the existence of wandering triangles important because solving this problem would help understand the combinatorial structure of the family of all polynomials of the appropriate degree.
In this paper for a closed lamination on the unit circle invariant under $z\mapsto z^3$ (cubic lamination) we prove that if it has a wandering triangle then there must be two distinct recurrent critical points in the corresponding quotient space ("topological Julia set") $J$ with the same limit set coinciding with the limit set of any wandering vertex (wandering vertices in $J$ correspond to wandering gaps in the lamination).
Keywords: Julia set, wandering triangle., lamination, growing tree.
Mathematics Subject Classification: Primary: 37F10; Secondary: 37E2.
Citation: Alexander Blokh. Necessary conditions for the existence of wandering triangles for cubic laminations. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 13-34. doi: 10.3934/dcds.2005.13.13
[1]

Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929
[2]

François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262
[3]

Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure and Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211
[4]

Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993
[5]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751
[6]

Koh Katagata. On a certain kind of polynomials of degree 4 with disconnected Julia set. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 975-987. doi: 10.3934/dcds.2008.20.975
[7]

Volodymyr Nekrashevych. The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$. Journal of Modern Dynamics, 2012, 6 (3) : 327-375. doi: 10.3934/jmd.2012.6.327
[8]

Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583
[9]

Yu-Hao Liang, Wan-Rou Wu, Jonq Juang. Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173
[10]

Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1321-1329. doi: 10.3934/dcds.2016.36.1321
[11]

Luiz Felipe Nobili França. Partially hyperbolic sets with a dynamically minimal lamination. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2717-2729. doi: 10.3934/dcds.2018114
[12]

Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453
[13]

Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511
[14]

Song Shao, Xiangdong Ye. Non-wandering sets of the powers of maps of a star. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1175-1184. doi: 10.3934/dcds.2003.9.1175
[15]

Klara Stokes, Maria Bras-Amorós. Associating a numerical semigroup to the triangle-free configurations. Advances in Mathematics of Communications, 2011, 5 (2) : 351-371. doi: 10.3934/amc.2011.5.351
[16]

Frédéric Naud, Anke Pohl, Louis Soares. Fractal Weyl bounds and Hecke triangle groups. Electronic Research Announcements, 2019, 26: 24-35. doi: 10.3934/era.2019.26.003
[17]

Yang Shen, Jiazhong Yang. Hearing the shape of right triangle billiard tables. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5537-5549. doi: 10.3934/dcds.2021087
[18]

Koh Katagata. Quartic Julia sets including any two copies of quadratic Julia sets. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2103-2112. doi: 10.3934/dcds.2016.36.2103
[19]

Luiz Henrique de Figueiredo, Diego Nehab, Jorge Stolfi, João Batista S. de Oliveira. Rigorous bounds for polynomial Julia sets. Journal of Computational Dynamics, 2016, 3 (2) : 113-137. doi: 10.3934/jcd.2016006
[20]

Robert L. Devaney, Daniel M. Look. Buried Sierpinski curve Julia sets. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 1035-1046. doi: 10.3934/dcds.2005.13.1035

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy