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Invariant measures on stationary Bratteli diagrams

Published online by Cambridge University Press:  17 July 2009

S. BEZUGLYI ,
J. KWIATKOWSKI ,
K. MEDYNETS  and
B. SOLOMYAK
S. BEZUGLYI
Affiliation:
Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkov, Ukraine (email: [email protected], [email protected])
J. KWIATKOWSKI
Affiliation:
College of Economics and Computer Sciences, Barczewskiego 11, 10106 Olsztyn, Poland (email: [email protected])
K. MEDYNETS
Affiliation:
Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkov, Ukraine (email: [email protected], [email protected])
B. SOLOMYAK
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA (email: [email protected])
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Abstract

We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we give an explicit description of all ergodic probability measures that are invariant with respect to the tail equivalence relation (or the Vershik map); these measures are completely described by the incidence matrix of the Bratteli diagram. Since such diagrams correspond to substitution dynamical systems, our description provides an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 973 - 1007
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Akin, E.. Measures on Cantor space. Topology Proc. 24 (1999), 1–34.Google Scholar
[2]Akin, E.. Good measures on Cantor sets. Trans. Amer. Math. Soc. 357 (2004), 2681–2722.CrossRefGoogle Scholar
[3]Allouche, J.-P. and Shallit, J.. Automatic Sequences. Theory, Applications, Generalizations. Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
[4]Bailey, S., Keane, M., Petersen, K. and Salama, I.. Ergodicity of the adic transformation on the Euler graph. Math. Proc. Cambridge Philos. Soc. 141 (2006), 231–238.CrossRefGoogle Scholar
[5]Bauer, W. and Sigmund, K.. Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math. 79 (1975), 81–92.CrossRefGoogle Scholar
[6]Bezuglyi, S., Kwiatkowski, J. and Medynets, K.. Aperiodic substitutional systems and their Bratteli diagrams. Ergod. Th. & Dynam. Sys. (2009) to appear.CrossRefGoogle Scholar
[7]Bezuglyi, S., Dooley, A. H. and Kwiatkowski, J.. Topologies on the group of Borel automorphisms of a standard Borel space. Topol. Methods Nonlinear Anal. 27 (2006), 333–385.Google Scholar
[8]Bressaud, X., Durand, F. and Maass, A.. Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems. J. London Math. Soc. 72(2) (2005), 799–816.CrossRefGoogle Scholar
[9]Bowen, R. and Marcus, B.. Unique ergodicity for horocycle foliations. Israel J. Math. 26(1) (1977), 43–67.CrossRefGoogle Scholar
[10]Cornfeld, I., Sinai, Ya. and Fomin, S.. Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar
[11]Cortez, M. I., Durand, F., Host, B. and Maass, A.. Continuous and measurable eigenfunctions for linearly recurrent dynamical Cantor systems. J. London Math. Soc. 67(3) (2003), 790–804.CrossRefGoogle Scholar
[12]Damanik, D. and Lenz, D.. Substitution dynamical systems: characterization of linear repetitivity and applications. J. Math. Anal. Appl. 321 (2006), 766–780.CrossRefGoogle Scholar
[13]Dekking, F. M. and Keane, M.. Mixing properties of substitutions. Zeit. Wahr. 42 (1978), 23–33.CrossRefGoogle Scholar
[14]Denker, M. and Keane, M.. Almost topological dynamical systems. Israel J. Math. 34 (1979), 139–160.CrossRefGoogle Scholar
[15]Dougherty, R., Jackson, S. and Kechris, A.. The structure of hyperfinite Borel equivalence relations. Trans. Amer. Math. Soc. 341 (1994), 193–225.CrossRefGoogle Scholar
[16]Downarowicz, T.. The Choquet simplex of invariant measures for minimal flows. Israel J. Math. 74 (1991), 241–256.CrossRefGoogle Scholar
[17]Downarowicz, T.. Minimal models for noninvertible and not uniquely ergodic systems. Israel J. Math. 156 (2006), 93–110.CrossRefGoogle Scholar
[18]Durand, F.. A theorem of Cobham for non-primitive substitutions. Acta Arith. 104(3) (2002), 225–241.Google Scholar
[19]Durand, F., Host, B. and Skau, C.. Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergod. Th. & Dynam. Sys. 19 (1999), 953–993.CrossRefGoogle Scholar
[20]Effros, E. G.. Dimensions and C *-algebras (CBMS Regional Conference Series in Mathematics, 46). Conference Board of the Mathematical Sciences, Washington, DC, 1981.CrossRefGoogle Scholar
[21]Ferenczi, S., Mauduit, C. and Nogueira, A.. Substitution dynamical systems: algebraic characterization of eigenvalues. Ann. Sci. L’École Norm. Sup. 29 (1996), 519–533.CrossRefGoogle Scholar
[22]Ferenczi, S., Fisher, A. and Talet, M.. Minimality and unique ergodicity for adic transformations. Preprint.Google Scholar
[23]Fisher, A.. Integer Cantor sets and an order-two ergodic theorem. Ergod. Th. & Dynam. Sys. 13 (1992), 45–64.CrossRefGoogle Scholar
[24]Fisher, A.. Nonstationary mixing and the unique ergodicity of adic transformations. Preprint.Google Scholar
[25]Forrest, A.. K-groups associated with substitution minimal systems. Israel J. Math. 98 (1997), 101–139.CrossRefGoogle Scholar
[26]Gantmacher, F. R.. The Theory of Matrices. Chelsea, New York, 1959.Google Scholar
[27]Giordano, T., Putnam, I. and Skau, C.. Topological orbit equivalence and C *-crossed products. J. Reine Angew. Math. 469 (1995), 51–111.Google Scholar
[28]Giordano, T., Putnam, I. and Skau, C.. Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys. 24 (2004), 441–475.CrossRefGoogle Scholar
[29]Gjerde, R. and Johansen, Ø.. Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows. Ergod. Th. & Dynam. Sys. 20 (2000), 1687–1710.CrossRefGoogle Scholar
[30]Glasner, E. and Weiss, B.. Quasi-factors of zero-entropy systems. J. Amer. Math. Soc. 8 (1995), 665–686.Google Scholar
[31]Glasner, E. and Weiss, B.. Weak orbit equivalence of Cantor minimal systems. Internat. J. Math. 6 (1995), 559–579.CrossRefGoogle Scholar
[32]Glasner, E. and Weiss, B.. Kazhdan’s property T and the geometry of the collection of invariant measures. Geom. Funct. Anal. 7 (1997), 917–935.CrossRefGoogle Scholar
[33]Handelman, D.. Reducible topological Markov chains via K 0-theory and Ext. Contemp. Math. 10 (1982), 41–76.CrossRefGoogle Scholar
[34]Herman, R. H., Putnam, I. and Skau, C.. Ordered Bratteli diagrams, dimension groups, and topological dynamics. Internat. J. Math. 3 (1992), 827–864.Google Scholar
[35]Host, B.. Valeurs propres de systèmes dynamiques définis par de substitutions de longueur variable. Ergod. Th. & Dynam. Sys. 6 (1986), 529–540.CrossRefGoogle Scholar
[36]Keane, M.. Non-ergodic interval exchange transformations. Israel J. Math. 26(2) (1977), 188–196.CrossRefGoogle Scholar
[37]Kerov, S. and Vershik, A.. Locally semisimple algebras. Combinatorial theory and the K0-functor. Current Problems in Mathematics. Newest Results 26 (1985), 3–56. (Russian) Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985.Google Scholar
[38]Környei, I.. On a theorem of Pisot. Publ. Math. Debrecen 34(3–4) (1987), 169–179.CrossRefGoogle Scholar
[39]Lind, D. and Marcus, B.. Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[40]Livshits, A. N.. On the spectra of adic transformations of Markov compacta. Russian Math. Surveys 42(3) (1987), 222–223.CrossRefGoogle Scholar
[41]Livshits, A. N.. A sufficient condition for weak mixing of substitutions and stationary adic transformations. Math. Notes 44(6) (1988), 920–925.CrossRefGoogle Scholar
[42]Medynets, K.. Cantor aperiodic systems and Bratteli diagrams. C. R. Math. Acad. Sci. Paris 342 (2006), 43–46.CrossRefGoogle Scholar
[43]Mignosi, F. and Séébold, P.. If a DOL language is k-power free then it is circular. Automata, Languages and Programming (Lund, 1993) (Lecture Notes in Computer Science, 700). Springer, Berlin, 1993, pp. 507–518.CrossRefGoogle Scholar
[44]Pansiot, J.-J.. Complexité des facteurs des mots infinis engendrés par morphismes itéré. Automata, Languages and Programming (Antwerp, 1984) (Lecture Notes in Computer Science, 172). Springer, Berlin, 1984, pp. 380–389.Google Scholar
[45]Petersen, K.. Ergodic Theory. Cambridge University Press, Cambridge, 1983.CrossRefGoogle Scholar
[46]Petersen, K. and Schmidt, K.. Symmetric Gibbs measures. Trans. Amer. Math. Soc. 349(7) (1997), 2775–2811.CrossRefGoogle Scholar
[47]Pullman, N. J.. A geometric approach to the theory of non-negative matrices. Linear Algebra Appl. 4 (1971), 711–718.Google Scholar
[48]Queffelec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294). Springer, Berlin, 1987.Google Scholar
[49]Schneider, H.. The influence of the marked reduced graph of a non-negative matrix on the Jordan form and on related properties: a survey. Linear Algebra Appl. 84 (1986), 161–189.CrossRefGoogle Scholar
[50]Sigmund, K.. Affine transformations on the space of probability measures. Astérisque 51 (1978), 415–427Dynamical Systems III (Warsaw, 1977).Google Scholar
[51]Solomyak, B.. Dynamics of self-similar tilings. Ergod. Th. & Dynam. Sys. 17 (1997), 695–738. B. Solomyak. Corrections to ‘Dynamics of self-similar tilings’. Ergod. Th. & Dynam. Sys. 19 (1999), 1685.CrossRefGoogle Scholar
[52]Tam, B. and Schneider, H.. On the core of a cone-preserving map. Trans. Amer. Math. Soc. 343 (1994), 479–524.CrossRefGoogle Scholar
[53]Tam, B. and Schneider, H.. On the invariant faces associated with a cone-preserving map. Trans. Amer. Math. Soc. 353 (2001), 209–245.CrossRefGoogle Scholar
[54]Vershik, A.. Uniform algebraic approximation of shift and multiplication operators. Soviet Math. Dokl. 24 (1981), 97–100.Google Scholar
[55]Vershik, A.. A theorem on periodic Markov approximation in ergodic theory. Ergodic Theory and Related Topics. Akademie, Berlin, 1982, pp. 195–206.Google Scholar
[56]Vershik, A. and Livshits, A.. Adic models of ergodic transformations, spectral theory, substitutions, and related topics. Adv. Soviet Math. 9 (1992), 185–204.Google Scholar
[57]Victory, H. D. Jr. On nonnegative solutions to matrix equations. SIAM J. Algebraic Discrete Methods 6 (1985), 406–412.Google Scholar
[58]Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar
[59]Yuasa, H.. Invariant measures for the subshifts arising from non-primitive substitutions. J. D’Analyse Math. 102 (2007), 143–180.CrossRefGoogle Scholar