Fractal representation of the attractive lamination of an automorphism of the free group

 Access to full text

 Full (PDF)

1. S. Akiyama, Pisot numbers and greedy algorithm, Number theory (Eger, 1996) (1998), 9-21, de Gruyter, Berlin Zbl0919.11063MR1628829
2. S. Akiyama, Self affine tiling and Pisot numeration system, Number theory and its applications (Kyoto, 1997) 2 (1999), 7-17, Kluwer Acad. Publ., Dordrecht Zbl0999.11065MR1738803
3. S. Akiyama, Cubic Pisot units with finite beta expansions, Algebraic number theory and Diophantine analysis (Graz, 1998) (2000), 11-26, de Gruyter, Berlin Zbl1001.11038MR1770451
4. S. Akiyama, T. Sadahiro, A self-similar tiling generated by the minimal Pisot number, Proceedings of the 13th Czech and Slovak International Conference on Number Theory (Ostravice, 1997) 6 (1998), 9-26 Zbl1024.11066MR1822510
5. P. Arnoux, Échanges d’intervalles et flots sur les surfaces, Monog. Enseign. Math. 29 (1981), 5-38 Zbl0471.28014MR609891
6. P. Arnoux, V. Berthé, S. Ito, Discrete planes, ${\mathbb{Z}}^2$-actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier (Grenoble) 52 (2002), 305-349 Zbl1017.11006MR1906478
7. P. Arnoux, V. Berthé, A. Siegel, Two-dimensional iterated morphisms and discrete planes, Theoret. Comput. Sci. 319 (2004), 145-176 Zbl1068.37004MR2074952
8. P. Arnoux, M. Furukado, E. O. Harriss, S. Ito, Algebraic numbers and free group automorphisms, Preprint (2005) Zbl1254.37015
9. P. Arnoux, S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 181-207 Zbl1007.37001MR1838930
10. M. Barge, B. Diamond, Coincidence for substitutions of Pisot type, Bull. Soc. Math. France 130 (2002), 619-626 Zbl1028.37008MR1947456
11. M. Barge, J. Kwapisz, Geometric theory of unimodular Pisot substitution Zbl1152.37011MR2262174
12. V. Berthé, A. Siegel, Purely periodic $\beta$-expansions in the Pisot non-unit case, (2005) Zbl1197.11139
13. V. Berthé, A. Siegel, Tilings associated with beta-numeration and substitutions, INTEGERS (Electronic Journal of Combinatorial Number Theory) 5 (2005) Zbl1139.37008MR2191748
14. M. Bestvina, M. Feighn, M. Handel, Laminations, trees, and irreducible automorphisms of free groups, GAFA 7 (1997), 215-244 Zbl0884.57002MR1445386
15. M. Bestvina, M. Feighn, M. Handel, The Tits alternative for Out($F_n$), I: Dynamics of exponentially growing automorphisms, Ann. Math. 151 (2000), 517-623 Zbl0984.20025MR1765705
16. M. Bestvina, M. Handel, Train tracks for surface homeomorphisms, Topology 34 (1995), 109-140 Zbl0837.57010MR1308491
17. M. Betsvina, M. Handel, Train tracks and automorphisms of free groups, Ann. Math. 135 (1992), 1-51 Zbl0757.57004MR1147956
18. V. Canterini, A. Siegel, Automate des préfixes-suffixes associé à une substitution primitive, J. Théor. Nombres Bordeaux 13 (2001), 353-369 Zbl1071.37011MR1879663
19. V. Canterini, A. Siegel, Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc. 353 (2001), 5121-5144 Zbl1142.37302MR1852097
20. D. Cooper, Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111 (1987), 453-456 Zbl0628.20029MR916179
21. M. Coornaert, T. Delzant, A. Papadopoulos, Géométrie et théorie des groupes, 1441 (1990), Springer Verlag, Berlin Zbl0727.20018MR1075994
22. T. Coulbois, A. Hilion, M. Lustig, $ℝ$-trees and laminations for free groups, (2006) Zbl1197.20019
23. J.-M. Dumont, A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions, Theoret. Comput. Sci. 65 (1989), 153-169 Zbl0679.10010MR1020484
24. J.-M. Dumont, A. Thomas, Digital sum moments and substitutions, Acta Arith. 64 (1993), 205-225 Zbl0774.11041MR1225425
25. H. Ei, S. Ito, Tilings for some non-irreducible Pisot substitutions, Discrete Mathematics and Theoretical Computer Science 7 (2005), 81-122 Zbl1153.37323MR2164061
26. Sur les groupes hyperboliques d’après Mikhael Gromov, 83 (1990), GhysE.E., Boston Zbl0731.20025
27. M. Gromov, Hyperbolic groups, Essays in group theory 8 (1987), 75-263, Springer-Verlag Zbl0634.20015MR919829
28. C. Holton, L. Q. Zamboni, Geometric realizations of substitutions, Bull. Soc. Math. France 126 (1998), 149-179 Zbl0931.11004MR1675970
29. S. Ito, M. Kimura, On Rauzy fractal, Japan J. Indust. Appl. Math. 8 (1991), 461-486 Zbl0734.28010MR1137652
30. S. Ito, M. Ohtsuki, Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms, Tokyo J. Math. 16 (1993), 441-472 Zbl0805.11056MR1247666
31. S. Ito, H. Rao, Atomic surfaces, tilings and coincidence I. Irreducible case, (2006) Zbl1143.37013MR2254640
32. J. C. Lagarias, Y. Wang, Substitution Delone sets, Discrete Comput. Geom. 29 (2003), 175-209 Zbl1037.52017MR1957227
33. D. Lind, B. Marcus, An introduction to symbolic dynamics and coding, (1995), Cambridge University Press, Cambridge Zbl1106.37301MR1369092
34. M. Lothaire, Algebraic combinatorics on words, 90 (2002), Cambridge University Press Zbl1001.68093MR1905123
35. M. Lothaire, Applied combinatorics on words, 105 (2005), Cambridge University Press Zbl1133.68067MR2165687
36. W. Massey, Algebraic topology: an introduction, 56 (1984), Springer, New York Zbl0457.55001MR448331
37. R. D. Mauldin, S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), 811-829 Zbl0706.28007MR961615
38. A. Messaoudi, Propriétés arithmétiques et dynamiques du fractal de Rauzy, J. Théor. Nombres Bordeaux 10 (1998), 135-162 Zbl0918.11048MR1827290
39. A. Messaoudi, Frontière du fractal de Rauzy et système de numération complexe, Acta Arith. 95 (2000), 195-224 Zbl0968.28005MR1793161
40. B. Praggastis, Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc. 351 (1999), 3315-3349 Zbl0984.11008MR1615950
41. N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, 1794 (2002), Springer-Verlag, Berlin Zbl1014.11015MR1970385
42. M. Queffélec, Substitution dynamical systems—spectral analysis, (1987), Lecture Notes in Mathematics, 1294. Springer-Verlag, Berlin Zbl0642.28013MR924156
43. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147-178 Zbl0522.10032MR667748
44. Y. Sano, P. Arnoux, S. Ito, Higher dimensional extensions of substitutions and their dual maps, J. Anal. Math. 83 (2001), 183-206 Zbl0987.11013MR1828491
45. E. Seneta, Non-negative matrices and Markov chains, (1981), Springer-Verlag Zbl0471.60001MR2209438
46. A. Siegel, Représentation géométrique, combinatoire et arithmétique des substitutions de type Pisot, (2000) 
47. A. Siegel, Représentation des systèmes dynamiques substitutifs non unimodulaires, Ergodic Theory Dynam. Systems 23 (2003), 1247-1273 Zbl1052.37009MR1997975
48. A. Siegel, Pure discrete spectrum dynamical system and periodic tiling associated with a substitution, Ann. Inst. Fourier (Grenoble) 54 (2004), 288-299 Zbl1083.37009MR2073838
49. V. F. Sirvent, Y. Wang, Self-affine tiling via substitution dynamical systems and Rauzy fractals, Pacific J. Math. 206 (2002), 465-485 Zbl1048.37015MR1926787
50. B. Tan, Z.-X. Wen, Y. Zhang, The structure of invertible substitutions on a three-letter alphabet, Advances in Applied Mathematics 32 (2004), 736-753 Zbl1082.68092MR2053843
51. W. P. Thurston, Groups, tilings and finite state automata, Lectures notes distributed in conjunction with the Colloquium Series (1989) 

1. Yann Jullian, Construction du cœur compact d’un arbre réel par substitution d’arbre
2. Guy Barat, Valérie Berthé, Pierre Liardet, Jörg Thuswaldner, Dynamical directions in numeration