American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function
  • CPAA Home
  • This Issue
  • Next Article
    Compact kernel sections for dissipative non-autonomous Zakharov equation on infinite lattices
January  2010, 9(1): 211-231. doi: 10.3934/cpaa.2010.9.211

Laplacians on the basilica Julia set

Luke G. Rogers 1, and  Alexander Teplyaev 1,

1. 

Department of Mathematics, University of Connecticut, Storrs CT 06269-3009, United States, United States

Received  January 2009 Revised  May 2009 Published  October 2009

We consider the basilica Julia set of the polynomial $P(z)=z^{2}-1$ and construct all possible resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the effective resistance metric coincides with the usual topology. Then we concentrate on two particular cases. One is a self-similar harmonic structure, for which the energy renormalization factor is $2$, the exponent in the Weyl law is $\log9/\log6$, and we can compute all the eigenvalues and eigenfunctions by a spectral decimation method. The other is graph-directed self-similar under the map $z\mapsto P(z)$; it has energy renormalization factor $\sqrt2$ and Weyl exponent $4/3$, but the exact computation of the spectrum is difficult. The latter Dirichlet form and Laplacian are in a sense conformally invariant on the basilica Julia set.
Keywords: Dirichlet form, spectral decimation., Fractal, Resistance form, eigenvalues, self-similarity, Julia set, Laplacian, eigenfunctions.
Mathematics Subject Classification: Primary: 28A80; Secondary: 37F50, 31C2.
Citation: Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure & Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211
[1]

Rogelio Valdez. Self-similarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 897-922. doi: 10.3934/dcds.2006.16.897
[2]

Hillel Furstenberg. From invariance to self-similarity: The work of Michael Hochman on fractal dimension and its aftermath. Journal of Modern Dynamics, 2019, 15: 437-449. doi: 10.3934/jmd.2019027
[3]

José Ignacio Alvarez-Hamelin, Luca Dall'Asta, Alain Barrat, Alessandro Vespignani. K-core decomposition of Internet graphs: hierarchies, self-similarity and measurement biases. Networks & Heterogeneous Media, 2008, 3 (2) : 371-393. doi: 10.3934/nhm.2008.3.371
[4]

Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
[5]

Selma Yildirim Yolcu, Türkay Yolcu. Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2209-2225. doi: 10.3934/dcds.2015.35.2209
[6]

Monique Dauge, Thomas Ourmières-Bonafos, Nicolas Raymond. Spectral asymptotics of the Dirichlet Laplacian in a conical layer. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1239-1258. doi: 10.3934/cpaa.2015.14.1239
[7]

Karthikeyan Rajagopal, Serdar Cicek, Akif Akgul, Sajad Jafari, Anitha Karthikeyan. Chaotic cuttlesh: king of camouage with self-excited and hidden flows, its fractional-order form and communication designs with fractional form. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1001-1013. doi: 10.3934/dcdsb.2019205
[8]

Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1
[9]

David Iglesias-Ponte, Juan Carlos Marrero, David Martín de Diego, Edith Padrón. Discrete dynamics in implicit form. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1117-1135. doi: 10.3934/dcds.2013.33.1117
[10]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 139-169. doi: 10.3934/naco.2017010
[11]

Abbas Bahri. Recent results in contact form geometry. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 21-30. doi: 10.3934/dcds.2004.10.21
[12]

Bernd Kawohl, Friedemann Schuricht. First eigenfunctions of the 1-Laplacian are viscosity solutions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 329-339. doi: 10.3934/cpaa.2015.14.329
[13]

Guojing Zhang, Steve Rosencrans, Xuefeng Wang, Kaijun Zhang. Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1061-1079. doi: 10.3934/dcds.2009.25.1061
[14]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[15]

Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 75-90. doi: 10.3934/dcds.2018004
[16]

Gary Lieberman. Nonlocal problems for quasilinear parabolic equations in divergence form. Conference Publications, 2003, 2003 (Special) : 563-570. doi: 10.3934/proc.2003.2003.563
[17]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
[18]

Emmanuel Hebey, Jérôme Vétois. Multiple solutions for critical elliptic systems in potential form. Communications on Pure & Applied Analysis, 2008, 7 (3) : 715-741. doi: 10.3934/cpaa.2008.7.715
[19]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[20]

Tony Lyons. Geophysical internal equatorial waves of extreme form. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4471-4486. doi: 10.3934/dcds.2019183

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences