Finite rank Bratteli diagrams: Structure of invariant measures
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- by S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak PDF
- Trans. Amer. Math. Soc. 365 (2013), 2637-2679 Request permission
Abstract:
We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or “regular” infinite) is obtained by an extension from a simple subdiagram. We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. A number of sufficient conditions for unique ergodicity are obtained. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. Several examples illustrate the broad range of possible behavior of finite rank diagrams and invariant measures on them. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. On the other hand, for the so-called “consecutive” ordering, the Vershik map is not strongly mixing on all finite rank diagrams.References
- Terrence M. Adams, Smorodinsky’s conjecture on rank-one mixing, Proc. Amer. Math. Soc. 126 (1998), no. 3, 739–744. MR 1443143, DOI 10.1090/S0002-9939-98-04082-9
- Sarah Bailey, Dynamical properties of some non-stationary, non-simple Bratteli-Vershik systems, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–The University of North Carolina at Chapel Hill. MR 2708436
- Sarah Bailey Frick and Karl Petersen, Random permutations and unique fully supported ergodicity for the Euler adic transformation, Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 5, 876–885 (English, with English and French summaries). MR 2453848, DOI 10.1214/07-AIHP133
- S. Bezuglyi, J. Kwiatkowski, and K. Medynets, Aperiodic substitution systems and their Bratteli diagrams, Ergodic Theory Dynam. Systems 29 (2009), no. 1, 37–72. MR 2470626, DOI 10.1017/S0143385708000230
- S. Bezuglyi, J. Kwiatkowski, K. Medynets, and B. Solomyak, Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Systems 30 (2010), no. 4, 973–1007. MR 2669408, DOI 10.1017/S0143385709000443
- Garrett Birkhoff, Extensions of Jentzsch’s theorem, Trans. Amer. Math. Soc. 85 (1957), 219–227. MR 87058, DOI 10.1090/S0002-9947-1957-0087058-6
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
- Michael D. Boshernitzan, A condition for unique ergodicity of minimal symbolic flows, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 425–428. MR 1182655, DOI 10.1017/S0143385700006866
- Michael D. Boshernitzan, Quantitative recurrence results, Invent. Math. 113 (1993), no. 3, 617–631. MR 1231839, DOI 10.1007/BF01244320
- Xavier Bressaud, Fabien Durand, and Alejandro Maass, On the eigenvalues of finite rank Bratteli-Vershik dynamical systems, Ergodic Theory Dynam. Systems 30 (2010), no. 3, 639–664. MR 2643706, DOI 10.1017/S0143385709000236
- Maria Isabel Cortez, Fabien Durand, Bernard Host, and Alejandro Maass, Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems, J. London Math. Soc. (2) 67 (2003), no. 3, 790–804. MR 1967706, DOI 10.1112/S0024610703004320
- David Damanik and Daniel Lenz, A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem, Duke Math. J. 133 (2006), no. 1, 95–123. MR 2219271, DOI 10.1215/S0012-7094-06-13314-8
- P. Dartnell, F. Durand, and A. Maass, Orbit equivalence and Kakutani equivalence with Sturmian subshifts, Studia Math. 142 (2000), no. 1, 25–45. MR 1792287, DOI 10.4064/sm-142-1-25-45
- F. M. Dekking and M. Keane, Mixing properties of substitutions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42 (1978), no. 1, 23–33. MR 466485, DOI 10.1007/BF00534205
- Tomasz Downarowicz, Survey of odometers and Toeplitz flows, Algebraic and topological dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., Providence, RI, 2005, pp. 7–37. MR 2180227, DOI 10.1090/conm/385/07188
- Tomasz Downarowicz and Alejandro Maass, Finite-rank Bratteli-Vershik diagrams are expansive, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 739–747. MR 2422014, DOI 10.1017/S0143385707000673
- Fabien Durand, Corrigendum and addendum to: “Linearly recurrent subshifts have a finite number of non-periodic subshift factors” [Ergodic Theory Dynam. Systems 20 (2000), no. 4, 1061–1078; MR1779393 (2001m:37022)], Ergodic Theory Dynam. Systems 23 (2003), no. 2, 663–669. MR 1972245, DOI 10.1017/S0143385702001293
- Fabien Durand, Combinatorics on Bratteli diagrams and dynamical systems, Combinatorics, automata and number theory, Encyclopedia Math. Appl., vol. 135, Cambridge Univ. Press, Cambridge, 2010, pp. 324–372. MR 2759109
- F. Durand, B. Host, and C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems 19 (1999), no. 4, 953–993. MR 1709427, DOI 10.1017/S0143385799133947
- Edward G. Effros and Chao Liang Shen, Dimension groups and finite difference equations, J. Operator Theory 2 (1979), no. 2, 215–231. MR 559606
- Edward G. Effros and Chao Liang Shen, The geometry of finite rank dimension groups, Illinois J. Math. 25 (1981), no. 1, 27–38. MR 602892
- Sébastien Ferenczi, Rank and symbolic complexity, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 663–682. MR 1406427, DOI 10.1017/S0143385700009032
- Sébastien Ferenczi, Systems of finite rank, Colloq. Math. 73 (1997), no. 1, 35–65. MR 1436950, DOI 10.4064/cm-73-1-35-65
- Sebastien Ferenczi, Albert M. Fisher, and Marina Talet, Minimality and unique ergodicity for adic transformations, J. Anal. Math. 109 (2009), 1–31. MR 2585390, DOI 10.1007/s11854-009-0027-y
- Albert M. Fisher, Nonstationary mixing and the unique ergodicity of adic transformations, Stoch. Dyn. 9 (2009), no. 3, 335–391. MR 2566907, DOI 10.1142/S0219493709002701
- Stefano Galatolo and Dong Han Kim, The dynamical Borel-Cantelli lemma and the waiting time problems, Indag. Math. (N.S.) 18 (2007), no. 3, 421–434. MR 2373690, DOI 10.1016/S0019-3577(07)80031-0
- Richard Gjerde and Ørjan Johansen, Bratteli-Vershik models for Cantor minimal systems associated to interval exchange transformations, Math. Scand. 90 (2002), no. 1, 87–100. MR 1887096, DOI 10.7146/math.scand.a-14363
- Thierry Giordano, Ian F. Putnam, and Christian F. Skau, Topological orbit equivalence and $C^*$-crossed products, J. Reine Angew. Math. 469 (1995), 51–111. MR 1363826
- K. R. Goodearl and D. E. Handelman, Stenosis in dimension groups and AF $C^{\ast }$-algebras, J. Reine Angew. Math. 332 (1982), 1–98. MR 656856, DOI 10.1515/crll.1982.332.1
- J. Hajnal, On products of non-negative matrices, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 3, 521–530. MR 396628, DOI 10.1017/S030500410005252X
- David E. Handelman, Eigenvectors and ratio limit theorems for Markov chains and their relatives, J. Anal. Math. 78 (1999), 61–116. MR 1714461, DOI 10.1007/BF02791129
- Darald J. Hartfiel, Nonhomogeneous matrix products, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. MR 1878339
- Richard H. Herman, Ian F. Putnam, and Christian F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827–864. MR 1194074, DOI 10.1142/S0129167X92000382
- Charles R. Johnson and Rafael Bru, The spectral radius of a product of nonnegative matrices, Linear Algebra Appl. 141 (1990), 227–240. MR 1076115, DOI 10.1016/0024-3795(90)90320-C
- Anatole Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math. 35 (1980), no. 4, 301–310. MR 594335, DOI 10.1007/BF02760655
- M. Keane, Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10 (1968), 335–353. MR 239047, DOI 10.1007/BF00531855
- John C. Martin, The structure of generalized Morse minimal sets on $n$ symbols, Trans. Amer. Math. Soc. 232 (1977), 343–355. MR 463400, DOI 10.1090/S0002-9947-1977-0463400-1
- Konstantin Medynets, Cantor aperiodic systems and Bratteli diagrams, C. R. Math. Acad. Sci. Paris 342 (2006), no. 1, 43–46 (English, with English and French summaries). MR 2193394, DOI 10.1016/j.crma.2005.10.024
- Xavier Méla, A class of nonstationary adic transformations, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 1, 103–123 (English, with English and French summaries). MR 2196974, DOI 10.1016/j.anihpb.2005.02.002
- N. J. Pullman, A geometric approach to the theory of nonnegative matrices, Linear Algebra Appl. 4 (1971), 297–312. MR 286816, DOI 10.1016/0024-3795(71)90001-2
- A. Rosenthal. Les systèmes de rang fini exact ne sont pas mélangeants. Preprint, 1984.
- Petar K. Rusev, Hermite functions of second kind, Serdica 2 (1976), no. 2, 177–190. MR 427707
- E. Seneta, Non-negative matrices and Markov chains, Springer Series in Statistics, Springer, New York, 2006. Revised reprint of the second (1981) edition [Springer-Verlag, New York; MR0719544]. MR 2209438
- A. M. Vershik and S. V. Kerov, Asymptotic theory of the characters of a symmetric group, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 15–27, 96 (Russian). MR 639197
- Krzysztof Wargan, S-adic dynamical systems and Bratteli diagrams, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–The George Washington University. MR 2702703
Additional Information
- S. Bezuglyi
- Affiliation: Institute for Low Temperature Physics, National Academy of Sciences of Ukraine, Kharkov, Ukraine
- MR Author ID: 215325
- Email: bezuglyi@ilt.kharkov.ua
- J. Kwiatkowski
- Affiliation: Department of Mathematics, University of Warmia and Mazury, 10-719 Olsztyn, Poland
- Email: jkwiat@mat.uni.torun.pl
- K. Medynets
- Affiliation: Department of Mathematics, United States Naval Academy, Annapolis, Maryland 21402
- MR Author ID: 752184
- Email: medynets@usna.edu
- B. Solomyak
- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 209793
- Email: solomyak@math.washington.edu
- Received by editor(s): December 23, 2010
- Received by editor(s) in revised form: September 19, 2011
- Published electronically: November 7, 2012
- Additional Notes: The research of the second author was supported by grant MNiSzW N N201384834.
The fourth author was supported in part by NSF grants DMS-0654408 and DMS-0968879. - © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 2637-2679
- MSC (2010): Primary 37B05, 37A25, 37A20
- DOI: https://doi.org/10.1090/S0002-9947-2012-05744-8
- MathSciNet review: 3020111