Invariant measures on Bratteli diagrams
Information from scientific report at the meeting of Presidium of NAS of Ukraine, June 17, 2015
DOI:
https://doi.org/10.15407/visn2015.09.080Keywords:
Cantor dynamical system, ergodic invariant measure, Bratteli diagram, orbit equivalenceAbstract
The results, stated in the report, are devoted to the problem of classification of Cantor dynamical systems up to orbit equivalence. The study of invariant measures for such dynamical systems plays the crucial role in the classification. The complete classification of the wide class of both finite and infinite Borel measures on Cantor spaces is given. In particular, the complete classification of ergodic invariant measures for aperiodic substitution dynamical systems is obtained. Such measures are invariant for the cofinal equivalence relation on the path spaces of stationary Bratteli diagrams. The structure of ergodic measures on the path space of an arbitrary Bratteli diagram is studied, in particular, the description of subdiagrams supporting finite ergodic invariant measures is found. These results are essential for deciding an open question about the classification of Cantor aperiodic dynamical systems up to orbit equivalence.
References
Morse M. A one-to-one representation of geodesics on a surface of negative curvature. Am. J. Math. 1921. 43: 33. http://doi.org/10.2307/2370306
Morse M., Hedlund G. Symbolic dynamics. Am. J. Math. 1938. 60(4): 815. http://doi.org/10.2307/2371264
Durand F. Combinatorics on Bratteli diagrams and dynamical systems. In: Combinatorics, Automata and Number Theory (Eds. V. Berthe, M. Rigo). (Cambridge: University Press, 2010). http://doi.org/10.1017/CBO9780511777653.007
Putnam I. Orbit equivalence of Cantor minimal systems: a survey and a new proof. Expo. Math. 2010. 28(2): 101. http://doi.org/10.1016/j.exmath.2009.06.002
Bezuglyi S., Karpel O. Bratteli diagrams: structure, measures, dynamics. arXiv:1503.03360.
Bratteli O. Inductive limits of finite-dimensional C*-algebras. Trans. Am. Math. Soc. 1972. 171: 195.
Vershik A.M. A theorem on Markov periodic approximation іn ergodic theory. Zap. Nauchn. Sem. LOMI. 1982. 115: 72. [in Russian].
Herman R.H., Putnam I., Skau C. Ordered Bratteli diagrams, dimension groups, and topological dynamics. Int. J. Math. 1992. 3(6): 827. http://doi.org/10.1142/S0129167X92000382
Medynets K. Cantor aperiodic systems and Bratteli diagrams. Comptes Rendus Mathematique. 2006. 342(1): 43. http://doi.org/10.1016/j.crma.2005.10.024
Giordano T., Putnam I., Skau C. Topological orbit equivalence and C*-crossed products. J. Reine Angew. Math. 1995. 469: 51.
Bezuglyi S., Karpel O. Homeomorphic Measures on Stationary Bratteli Diagrams. J. Funct. Anal. 2011. 261(12): 3519. http://ddoi.org/10.1016/j.jfa.2011.08.009
Karpel O. Infinite measures on Cantor spaces. J. Difference Equ. Appl. 2012. 18(4): 703. http://doi.org/10.1080/10236198.2011.620955
Durand F., Host B., Skau C. Substitutional dynamical systems. Bratteli diagrams and dimension groups. Ergodic Theory Dynam. Syst. 1999. 19(4): 953. http://doi.org/10.1017/S0143385799133947
Bezuglyi S., Kwiatkowski J., Medynets K. Aperiodic substitution systems and their Bratteli diagrams. Ergodic Theory Dynam. Syst. 2009. 29(1): 37. http://doi.org/10.1017/S0143385708000230
Oxtoby J.C., Ulam S.M. Measure preserving homeomorphisms and metrical transitivity. Ann. Math. 1941. 42: 874. http://doi.org/10.2307/1968772
Alpern S., Prasad V.S. Typical Dynamics of Volume Preserving Homeomorphisms. (Cambridge: Cambridge University Press, 2000).
Akin E. Measures on Cantor space. Topology Proc. 1999. 24: 1.
Navarro-Bermudez F.J. Topologically equivalent measures in the Cantor space. Proc. Am. Math. Soc. 1979. 77(2): 229. http://doi.org/10.2307/2042644
Akin E., Dougherty R., Mauldin R.D., Yingst A. Which Bernoulli measures are good measures? Colloq. Math. 2008. 110: 243. http://doi.org/10.4064/cm110-2-2
Austin T.D. A pair of non-homeomorphic product measures on the Cantor set. Math. Proc. Cam. Phil. Soc. 2007. 142: 103. http://doi.org/10.1017/S0305004106009741
Karpel O. Good measures on locally compact Cantor sets. J. Math. Phys. Anal. Geom. 2012. 8: 260.
Bezuglyi S., Karpel O. Orbit Equivalent Substitution Dynamical Systems and Complexity. Proc. Am. Math. Soc. 2014. 142: 4155. http://doi.org/10.1090/S0002-9939-2014-12139-3
Bezuglyi S., Karpel O., Kwiatkowski J. Subdiagrams of Bratteli diagrams supporting finite invariant measures. J. Math. Phys. Anal. Geom. 2015. 11(1): 3.
Adamska M., Bezuglyi S., Karpel O., Kwiatkowski J. Subdiagrams and invariant measures of Bratteli diagrams. arXiv:1502.05690.
Bezuglyi S., Jorgensen P. Representations of Cuntz-Krieger relations, dynamics on Bratteli diagrams, and path-space measures. arXiv:1410.2318.
Bezuglyi S., Handelman D. Measures on Cantor sets: The good, the ugly, the bad. Trans. Am. Math. Soc. 2014. 366(12): 6247. http://doi.org/10.1090/S0002-9947-2014-06035-2
