In this paper, we define some Markov chains associated with Vershik maps on Bratteli diagrams. We study probabilistic and spectral properties of their transition operators and we prove that the spectra of these operators are connected to Julia sets in higher dimensions. We also study topological properties of these spectra.
Dans cet article, nous définissons des chaînes de Markov associées aux applications de Vershik sur des diagrammes de Bratteli. Nous étudions des propriétés probabilistes et spectrales de leurs opérateurs de transition et nous prouvons que les spectres de ces opérateurs sont liés à des ensembles de Julia de dimensions supérieures. Nous étudions également des propriétés topologiques de ces spectres.
Revised:
Accepted:
Published online:
Keywords: Markov chains, stochastic Vershik map, Bratteli diagrams, spectrum of transition operators, fibered Julia sets
Mot clés : chaînes de Markov, application de Vershik stochastique, diagrammes de Bratteli, spectre des opérateurs de transition, ensembles de Julia fibrés
Caprio, Danilo A. 1; Messaoudi, Ali 1; Valle, Glauco 2
@article{AIF_2020__70_6_2543_0, author = {Caprio, Danilo A. and Messaoudi, Ali and Valle, Glauco}, title = {Stochastic adding machines based on {Bratteli} diagrams}, journal = {Annales de l'Institut Fourier}, pages = {2543--2581}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {6}, year = {2020}, doi = {10.5802/aif.3364}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3364/} }
TY - JOUR AU - Caprio, Danilo A. AU - Messaoudi, Ali AU - Valle, Glauco TI - Stochastic adding machines based on Bratteli diagrams JO - Annales de l'Institut Fourier PY - 2020 SP - 2543 EP - 2581 VL - 70 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3364/ DO - 10.5802/aif.3364 LA - en ID - AIF_2020__70_6_2543_0 ER -
%0 Journal Article %A Caprio, Danilo A. %A Messaoudi, Ali %A Valle, Glauco %T Stochastic adding machines based on Bratteli diagrams %J Annales de l'Institut Fourier %D 2020 %P 2543-2581 %V 70 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3364/ %R 10.5802/aif.3364 %G en %F AIF_2020__70_6_2543_0
Caprio, Danilo A.; Messaoudi, Ali; Valle, Glauco. Stochastic adding machines based on Bratteli diagrams. Annales de l'Institut Fourier, Volume 70 (2020) no. 6, pp. 2543-2581. doi : 10.5802/aif.3364. https://aif.centre-mersenne.org/articles/10.5802/aif.3364/
[1] On the spectrum of stochastic perturbations of the shift and Julia sets, Fundam. Math., Volume 218 (2012) no. 1, pp. 47-68 | DOI | MR | Zbl
[2] Dynamics of linear operators, Cambridge Tracts in Mathematics, 179, Cambridge University Press, 2009, xiv+337 pages | DOI | MR | Zbl
[3] Topologies on the group of Borel automorphisms of a standard Borel space, Topol. Methods Nonlinear Anal., Volume 27 (2006) no. 2, pp. 333-385 | MR | Zbl
[4] Bratteli diagrams: structure, measures, dynamics, Dynamics and numbers (Contemporary Mathematics), Volume 669, American Mathematical Society, 2016, pp. 1-36 | DOI | MR | Zbl
[5] Inductive limits of finite dimensional -algebras, Trans. Am. Math. Soc., Volume 171 (1972), pp. 195-234 | DOI | MR | Zbl
[6] Markov chains. Gibbs fields, Monte Carlo simulation, and queues, Texts in Applied Mathematics, 31, Springer, 1999, xviii+444 pages | DOI | MR | Zbl
[7] Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems, J. Lond. Math. Soc., Volume 72 (2005) no. 3, pp. 799-816 | DOI | MR | Zbl
[8] Quadratic Julia sets with positive area, Ann. Math., Volume 176 (2012) no. 2, pp. 673-746 | DOI | MR | Zbl
[9] A class of adding machines and Julia sets, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 11, pp. 5951-5970 | DOI | MR | Zbl
[10] Complex dynamics, Universitext: Tracts in Mathematics, Springer, 1993, x+175 pages | DOI | MR | Zbl
[11] An introduction to chaotic dynamical systems, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Group, 1989, xviii+336 pages | MR | Zbl
[12] Disques de Siegel et anneaux de Herman, Séminaire Bourbaki, Vol. 1986/87 (Astérisque), Société Mathématique de France, 1988 no. 152-153, pp. 151-172 | MR | Zbl
[13] Étude dynamique des polynômes complexes. Partie I, Publications Mathématiques d’Orsay, 84, Université de Paris-Sud, 1984, 75 pages | MR | Zbl
[14] Étude dynamique des polynômes complexes. Partie II, Publications Mathématiques d’Orsay, 85, Université de Paris-Sud, 1985, v+154 pages (With the collaboration of P. Lavaurs, Tan Lei and P. Sentenac) | MR
[15] Combinatorics on Bratteli diagrams and dynamical systems, Combinatorics, automata and number theory (Encyclopedia of Mathematics and Its Applications), Volume 135, Cambridge University Press, 2010, pp. 324-372 | DOI | MR | Zbl
[16] Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, 1990, xxii+288 pages | MR | Zbl
[17] Sur les équations fonctionnelles, Bull. Soc. Math. Fr., Volume 47 (1919), pp. 161-271 | DOI | Numdam | MR | Zbl
[18] Sur les équations fonctionnelles, Bull. Soc. Math. Fr., Volume 48 (1920), pp. 33-94 | DOI | Numdam | MR
[19] Fatou and Julia sets for entire mappings in , Math. Ann., Volume 311 (1998) no. 1, pp. 27-40 | DOI | MR | Zbl
[20] Topological orbit equivalence and -crossed products, J. Reine Angew. Math., Volume 469 (1995), pp. 51-111 | MR | Zbl
[21] Linear chaos, Universitext, Springer, 2011, xii+386 pages | DOI | MR | Zbl
[22] Dynamics of polynomial automorphisms of , Ark. Mat., Volume 40 (2002) no. 2, pp. 207-243 | DOI | MR | Zbl
[23] Ordered Bratteli diagrams, dimension groups and topological dynamics, Int. J. Math., Volume 3 (1992) no. 6, pp. 827-864 | DOI | MR | Zbl
[24] Substitution subshifts and Bratteli diagrams, Topics in symbolic dynamics and applications (Temuco, 1997) (London Mathematical Society Lecture Note Series), Volume 279, Cambridge University Press, 2000, pp. 35-55 | MR | Zbl
[25] Mémoire sur la permutabilité des fractions rationnelles, Ann. Sci. Éc. Norm. Supér., Volume 39 (1922), pp. 131-215 | DOI | Numdam | MR | Zbl
[26] Julia sets, Chaos and fractals. The mathematics behind the computer graphics (Providence, RI, 1988) (Proceedings of Symposia in Applied Mathematics), Volume 39, American Mathematical Society, 1989, pp. 57-74 | DOI | MR
[27] A stochastic adding machine and complex dynamics, Nonlinearity, Volume 13 (2000) no. 6, pp. 1889-1903 | DOI | MR | Zbl
[28] Dynamics of rational transformations: topological picture, Usp. Mat. Nauk, Volume 41 (1986) no. 4(250), pp. 35-95 | MR
[29] Cantor aperiodic systems and Bratteli diagrams, C. R. Math. Acad. Sci. Paris, Volume 342 (2006) no. 1, pp. 43-46 | DOI | MR | Zbl
[30] Spectrum of stochastic adding machines and fibered Julia sets, Stoch. Dyn., Volume 13 (2013) no. 3, 1250021, 26 pages | DOI | MR | Zbl
[31] Eigenvalues of Fibonacci stochastic adding machine, Stoch. Dyn., Volume 10 (2010) no. 2, pp. 291-313 | DOI | MR | Zbl
[32] Stochastic adding machine and -dimensional Julia sets, Discrete Contin. Dyn. Syst., Volume 34 (2014) no. 12, pp. 5247-5269 | DOI | MR | Zbl
[33] Spectra of generalized stochastic adding machines, Fundam. Math., Volume 241 (2018) no. 1, pp. 17-43 | DOI | MR | Zbl
[34] On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87) (Lecture Notes in Mathematics), Volume 1342, Springer, 1988, pp. 465-563 | DOI | MR | Zbl
[35] Hyperbolicité des polynômes fibrés, Bull. Soc. Math. Fr., Volume 127 (1999) no. 3, pp. 393-428 | DOI | Numdam | MR | Zbl
[36] Uniform algebraic approximation of shift and multiplication operators, Dokl. Akad. Nauk SSSR, Volume 259 (1981) no. 3, pp. 526-529 | MR | Zbl
[37] A theorem on Markov periodic approximation in ergodic theory, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, Volume 115 (1982), pp. 72-82 | MR
[38] Linéarisation des germes de difféomorphismes holomorphes de , C. R. Math. Acad. Sci. Paris, Volume 306 (1988) no. 1, pp. 55-58 | MR | Zbl
Cited by Sources: