 Previous Article
 JMD Home
 This Issue

Next Article
Unbounded orbits for outer billiards I
Selfsimilar groups, operator algebras and Schur complement
1.  Texas A&M University, College Station, Texas, United States, United States 
The second part deals with Schur complement transformations of elements of selfsimilar algebras. We study the properties of such transformations and apply them to the spectral problem for Markov type elements in selfsimilar $C$*algebras. This is related to the spectral problem of the discrete Laplace operator on groups and graphs. Application of the Schur complement method in many situations reduces the spectral problem to study of invariant sets (very often of the type of a "strange attractor'') of a multidimensional rational transformation. A number of illustrating examples is provided. Finally, we observe a relation between Schur complement transformations and BartholdiKaimanovichVirag transformations of random walks on selfsimilar groups.
[1] 
G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically selfsimilar dynamics. Conference Publications, 2005, 2005 (Special) : 131141. doi: 10.3934/proc.2005.2005.131 
[2] 
Christoph Bandt, Helena PeÑa. Polynomial approximation of selfsimilar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems  A, 2017, 37 (9) : 46114623. doi: 10.3934/dcds.2017198 
[3] 
Weronika Biedrzycka, Marta TyranKamińska. Selfsimilar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems  B, 2018, 23 (1) : 1327. doi: 10.3934/dcdsb.2018002 
[4] 
Marco Cannone, Grzegorz Karch. On selfsimilar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801808. doi: 10.3934/krm.2013.6.801 
[5] 
Rogelio Valdez. Selfsimilarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete & Continuous Dynamical Systems  A, 2006, 16 (4) : 897922. doi: 10.3934/dcds.2006.16.897 
[6] 
Hideo Kubo, Kotaro Tsugawa. Global solutions and selfsimilar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems  A, 2003, 9 (2) : 471482. doi: 10.3934/dcds.2003.9.471 
[7] 
K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. selfsimilar vanishing diffusion limits. Communications on Pure & Applied Analysis, 2002, 1 (1) : 5176. doi: 10.3934/cpaa.2002.1.51 
[8] 
Anna Chiara Lai, Paola Loreti. Selfsimilar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401419. doi: 10.3934/nhm.2015.10.401 
[9] 
D. G. Aronson. Selfsimilar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems  B, 2012, 17 (6) : 16851691. doi: 10.3934/dcdsb.2012.17.1685 
[10] 
Qiaolin He. Numerical simulation and selfsimilar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems  B, 2010, 13 (1) : 101116. doi: 10.3934/dcdsb.2010.13.101 
[11] 
F. Berezovskaya, G. Karev. Bifurcations of selfsimilar solutions of the FokkerPlank equations. Conference Publications, 2005, 2005 (Special) : 9199. doi: 10.3934/proc.2005.2005.91 
[12] 
Bendong Lou. Selfsimilar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857879. doi: 10.3934/nhm.2012.7.857 
[13] 
Shota Sato, Eiji Yanagida. Singular backward selfsimilar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems  S, 2011, 4 (4) : 897906. doi: 10.3934/dcdss.2011.4.897 
[14] 
Shota Sato, Eiji Yanagida. Forward selfsimilar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems  A, 2010, 26 (1) : 313331. doi: 10.3934/dcds.2010.26.313 
[15] 
L. Olsen. Rates of convergence towards the boundary of a selfsimilar set. Discrete & Continuous Dynamical Systems  A, 2007, 19 (4) : 799811. doi: 10.3934/dcds.2007.19.799 
[16] 
Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to selfsimilar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems  A, 2008, 21 (3) : 703716. doi: 10.3934/dcds.2008.21.703 
[17] 
Hyungjin Huh. Selfsimilar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 5360. doi: 10.3934/eect.2018003 
[18] 
Kin Ming Hui. Existence of selfsimilar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems  A, 2019, 39 (2) : 863880. doi: 10.3934/dcds.2019036 
[19] 
José Ignacio AlvarezHamelin, Luca Dall'Asta, Alain Barrat, Alessandro Vespignani. Kcore decomposition of Internet graphs: hierarchies, selfsimilarity and measurement biases. Networks & Heterogeneous Media, 2008, 3 (2) : 371393. doi: 10.3934/nhm.2008.3.371 
[20] 
Peter V. Gordon, Cyrill B. Muratov. Selfsimilarity and longtime behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767780. doi: 10.3934/nhm.2012.7.767 
2018 Impact Factor: 0.295
Tools
Metrics
Other articles
by authors
[Back to Top]