Symmetric and congruent Rauzy fractals

Sobre o artigo

co-autores Klaus Scheicher
Víctor F. Sirvent
idioma inglês
publicado no Indagationes Mathematicae 27, v. 3 (2016)
páginas 799 a 820
DOI 10.1016/j.indag.2016.01.011
suportado por FWF, projeto P23990
título em português Simetrias e congruências de fractais de Rauzy

Resumo

Dois fractais de Rauzy são congruentes se entre eles existe uma transformação afim. Nós damos condições a substituições de Pisot primitivos e unimodulares que garantem a congruência dos associados fractais de Rauzy. Usamos estes resultados para descrever uma família grande de substituções que produz fractais de Rauzy simétricos (referente a um ponto) em vista da linguagem induzida.

Bibliografia

  1. S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee, A. Siegel, On the Pisot substitution conjecture, in Mathematics of Aperiodic Order, vol. 309 of Progress in Mathematics, Birkhäuser, 2015, 33—72.
  2. S. Akiyama, J.-Y. Lee, Overlap coincidence to strong coincidence in substitution tiling dynamics, European J. Combin. 39 (2014), 233—243.
  3. J.-P. Allouche, M. Baake, J. Cassaigne, D.Damanik}, Palindrome complexity, Theoret. Comput. Sci. 292 (2003), 9—31. Selected papers in honor of Jean Berstel.
  4. J.-P. Allouche, T. Johnson, Narayana's cows and delayed morphisms, in 3èmes Journées d'Informatique Musicale (JIM '96), Ile de Tatihou, 1996, 2—7.
  5. P. Arnoux, S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 181—207, Journées Montoises d'Informatique Théorique, Marne-la-Vallée, 2000.
  6. P. Arnoux, G.~Rauzy, Représentation géométrique de suites de complexité 2n+1, Bull. Soc. Math. France 119 (1991), 199—215.
  7. L. Balková, E. Pelantová, Š. Starosta}, Infinite words with finite defect, Adv. in Appl. Math. 47 (2011), 562—574.
  8. G. Barat, V. Berthé, P. Liardet, J.M. Thuswaldner, Dynamical directions in numeration, Ann. Inst. Fourier (Grenoble) 56 (2006), 1987—2092.
  9. M. Barge, The Pisot Conjecture for β-substitutions, pre-publicação, disponível no arXiv.
  10. M. Barge, B. Diamond, Coincidence for substitutions of Pisot type, Bull. Soc. Math. France 130 (2002), 619—626.
  11. V. Berthé, T. Jolivet, A. Siegel, Substitutive Arnoux-Rauzy sequences have pure discrete spectrum, Unif. Distrib. Theory 7 (2012), 173—197.
  12. V. Berthé, T. Jolivet, A. Siegel, Connectedness of fractals associated with Arnoux-Rauzy substitutions, RAIRO Theor. Inform. Appl. 48 (2014), 249—266.
  13. V. Berthé, A. Siegel, Tilings associated with beta-numeration and substitutions, Integers 5 (2005), A2.
  14. V. Berthé, A. Siegel, J.M. Thuswaldner, Substitutions, Rauzy fractals and tilings, in Combinatorics, automata and number theory, vol. 135 of Encyclopedia Math. Appl., Cambridge Univ. Press, Cambridge, 2010, 248—323.
  15. J. Cassaigne, S. Ferenczi, A. Messaoudi, Weak mixing and eigenvalues for Arnoux-Rauzy sequences, Ann. Inst. Fourier (Grenoble) 58 (2008), 1983—2005.
  16. J. Cassaigne, S. Ferenczi, L.Q. Zamboni, Imbalances in Arnoux-Rauzy sequences, Ann. Inst. Fourier (Grenoble) 50 (2000), 1265—1276.
  17. N.P. Fogg, Substitutions in dynamics, arithmetics and combinatorics, vol. 1794 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002. Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel.
  18. T. Harju, J. Vesti, L.Q. Zamboni, On a question of Hof, Knill and Simon on palindromic substitutive systems, disponível no arXiv.
  19. A. Hof, O. Knill, B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Commun. Math. Phys. 174 (1995), 149—159.
  20. M. Hollander, B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum, Ergodic Theory Dynam. Systems. 23 (2003), 533—540.
  21. B. Host, Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable, Ergodic Theory Dynam. Systems 6 (1986), 529—540.
  22. S. Labbé, A counterexample to a question of Hof, Knill and Simon, Electron. J. Combin. 21 (2014), artigo n° #P3.11.
  23. B. Loridant, Topology of a class of cubic Rauzy fractals, Osaka J. Math., aceito.
  24. B. Loridant, A. Messaoudi, P. Surer, J.M. Thuswaldner, Tilings induced by a class of cubic Rauzy fractals, Theoret. Comput. Sci. 477 (2013), 6—31.
  25. B. Mossé, Puissances de mots et reconnaissabilité des points fixes d´une substitution, Theoret. Comput. Sci. 99 (1992), 327—334.
  26. M. Queffélec, Substitution dynamical systems. Spectral analysis. 2nd ed., Lecture Notes in Mathematics 1294. Dordrecht: Springer. xv, 351 p., 2010.
  27. J.L. Ramírez, V.F. Sirvent, On the k-Narayana sequence by matrix methods, Ann. Math. Inform., aceito.
  28. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147—178.
  29. T. Sellami, Geometry of the common dynamics of Pisot substitutions with the same incidence matrix, C. R. Math. Acad. Sci. Paris 348 (2010), 1005—1008.
  30. T. Sellami, V.F. Sirvent, Symmetric intersections of Rauzy fractals, Quaest. Math, aceito.
  31. A. Siegel, Pure discrete spectrum dynamical system and periodic tiling associated with a substitution, Ann. Inst. Fourier (Grenoble) 54 (2004), 341—381.
  32. A. Siegel, J.M. Thuswaldner, Topological properties of Rauzy fractals, Mém. Soc. Math. Fr. (N.S.) (2009).
  33. B. Sing, Pisot Substitutions and Beyond, tese de doutorado, Universität Bielefeld, 2006..
  34. B. Sing, V.F. Sirvent, Geometry of the common dynamics of flipped Pisot substitutions, Monatsh. Math. 155 (2008), 431—448.
  35. V.F. Sirvent, Geodesic laminations as geometric realizations of Arnoux-Rauzy sequences, Bull. Belg. Math. Soc. Simon Stevin 10 (2003), 221—229.
  36. V.F. Sirvent, Symmetries in k-bonacci adic systems, Integers 11B (2011), artigo n° A15, 18.
  37. V.F. Sirvent, Symmetries in Rauzy fractals, Unif. Distrib. Theory 7 (2012), 155—171.
  38. V.F. Sirvent, Y. Wang, Self-affine tiling via substitution dynamical systems and Rauzy fractals, Pacific J. Math. 206 (2002), 465—485.
  39. B. Tan, Mirror substitutions and palindromic sequences, Theoret. Comput. Sci. 389 (2007), 118—124.
  40. B. Tan, Z.-Y. Wen, Invertible substitutions and Sturmian sequences, European J. Combin. 24 (2003), 983—1002.
  41. J.M. Thuswaldner, Unimodular Pisot substitutions and their associated tiles, J. Théor. Nombres Bordeaux 18 (2006), 487—536.

Links

Indagationes Mathematicae
Fundação Austríaca de Ciência (FWF)