Periodic codings of Bratteli-Vershik systems
DOI:
https://doi.org/10.7146/math.scand.a-117570Abstract
We develop conditions for the coding of a Bratteli-Vershik system according to initial path segments to be periodic, equivalently for a constructive symbolic recursive scheme corresponding to a cutting and stacking process to produce a periodic sequence. This is a step toward understanding when a Bratteli-Vershik system can be essentially faithfully represented by means of a natural coding as a subshift on a finite alphabet.
References
Adams, T., Ferenczi, S., and Petersen, K., Constructive symbolic presentations of rank one measure-preserving systems, Colloq. Math. 150 (2017), no. 2, 243–255. https://doi.org/10.4064/cm7124-3-2017
Berthé, V., Steiner, W., Thuswaldner, J. M., and Yassawi, R., Recognizability for sequences of morphisms, Ergodic Theory Dynam. Systems 39 (2019), no. 11, 2896–2931. https://doi.org/10.1017/etds.2017.144
Bezuglyi, S. and Karpel, O., Bratteli diagrams: structure, measures, dynamics, in “Dynamics and numbers”, Contemp. Math., vol. 669, Amer. Math. Soc., Providence, RI, 2016, pp. 1–36. https://doi.org/10.1090/conm/669/13421
Bezuglyi, S., Kwiatkowski, J., and Medynets, K., Aperiodic substitution systems and their Bratteli diagrams, Ergodic Theory Dynam. Systems 29 (2009), no. 1, 37–72. https://doi.org/10.1017/S0143385708000230
Bezuglyi, S., Kwiatkowski, J., Medynets, K., and Solomyak, B., Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Systems 30 (2010), no. 4, 973–1007. https://doi.org/10.1017/S0143385709000443
Bezuglyi, S., Kwiatkowski, J., Medynets, K., and Solomyak, B., Finite rank Bratteli diagrams: structure of invariant measures, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2637–2679. https://doi.org/10.1090/S0002-9947-2012-05744-8
Dai, I., Garcia, X., Pădurariu, T., and Silva, C. E., On rationally ergodic and rationally weakly mixing rank-one transformations, Ergodic Theory Dynam. Systems 35 (2015), no. 4, 1141–1164. https://doi.org/10.1017/etds.2013.96
Danilenko, A. I., Strong orbit equivalence of locally compact Cantor minimal systems, Internat. J. Math. 12 (2001), no. 1, 113–123. https://doi.org/10.1142/S0129167X0100068X
Danilenko, A. I., Infinite rank one actions and nonsingular Chacon transformations, Illinois J. Math. 48 (2004), no. 3, 769–786.
Danilenko, A. I., Rank-one actions, their $(C,F)$-models and constructions with bounded parameters, eprint arxiv:1610.09851 [math.DS], 2016.
Downarowicz, T. and Maass, A., Finite-rank Bratteli-Vershik diagrams are expansive, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 739–747. https://doi.org/10.1017/S0143385707000673
Durand, F., Combinatorics on Bratteli diagrams and dynamical systems, in “Combinatorics, automata and number theory”, Encyclopedia Math. Appl., vol. 135, Cambridge Univ. Press, Cambridge, 2010, pp. 324–372.
Eigen, S., Hajian, A., Ito, Y., and Prasad, V., Weakly wandering sequences in ergodic theory, Springer Monographs in Mathematics, Springer, Tokyo, 2014. https://doi.org/10.1007/978-4-431-55108-9
El Abdalaoui, E. H., Lemańczyk, M., and de la Rue, T., On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal. 266 (2014), no. 1, 284–317. https://doi.org/10.1016/j.jfa.2013.09.005
Ferenczi, S., Systems of finite rank, Colloq. Math. 73 (1997), no. 1, 35–65. https://doi.org/10.4064/cm-73-1-35-65
Ferenczi, S. and Monteil, T., Infinite words with uniform frequencies, and invariant measures, in “Combinatorics, automata and number theory”, Encyclopedia Math. Appl., vol. 135, Cambridge Univ. Press, Cambridge, 2010, pp. 373–409.
Foreman, M. and Weiss, B., A characterization of odometer based systems: Working notes, preprint (personal communication), 2016.
Foreman, M. and Weiss, B., From odometers to circular systems: A global structure theorem, eprint arxiv:1703.07093 [math.DS], 2017.
Frick, S., Petersen, K., and Shields, S., Dynamical properties of some adic systems with arbitrary orderings, Ergodic Theory Dynam. Systems 37 (2017), no. 7, 2131–2162. https://doi.org/10.1017/etds.2015.128
Frick, S. B., Limited scope adic transformations, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), no. 2, 269–285. https://doi.org/10.3934/dcdss.2009.2.269
Herman, R. H., Putnam, I. F., and Skau, C. F., Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827–864. https://doi.org/10.1142/S0129167X92000382
Jewett, R. I., The prevalence of uniquely ergodic systems, J. Math. Mech. 19 (1969/1970), 717–729.
Kalikow, S. A., Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory Dynam. Systems 4 (1984), no. 2, 237–259. https://doi.org/10.1017/S014338570000242X
Krieger, W., On unique ergodicity, in “Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory”, 1972, pp. 327–346.
Lothaire, M., Applied combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 17, Addison-Wesley, Reading, Mass., 1983, a collective work by D. Perrin, et al.
Lothaire, M., Applied combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 105, Cambridge University Press, Cambridge, 2005, a collective work by J. Berstel, et al. https://doi.org/10.1017/CBO9781107341005
Medynets, K., Cantor aperiodic systems and Bratteli diagrams, C. R. Math. Acad. Sci. Paris 342 (2006), no. 1, 43–46. https://doi.org/10.1016/j.crma.2005.10.024
Méla, X., A class of nonstationary adic transformations, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 1, 103–123. https://doi.org/10.1016/j.anihpb.2005.02.002
Méla, X. and Petersen, K., Dynamical properties of the Pascal adic transformation, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 227–256. https://doi.org/10.1017/S0143385704000173
Mela, X. S., Dynamical properties of the Pascal adic and related systems, Ph.D. thesis, The University of North Carolina at Chapel Hill, 2002.
Yuasa, H., Invariant measures for the subshifts arising from non-primitive substitutions, J. Anal. Math. 102 (2007), 143–180. https://doi.org/10.1007/s11854-007-0019-8
Berthé, V., Steiner, W., Thuswaldner, J. M., and Yassawi, R., Recognizability for sequences of morphisms, Ergodic Theory Dynam. Systems 39 (2019), no. 11, 2896–2931. https://doi.org/10.1017/etds.2017.144
Bezuglyi, S. and Karpel, O., Bratteli diagrams: structure, measures, dynamics, in “Dynamics and numbers”, Contemp. Math., vol. 669, Amer. Math. Soc., Providence, RI, 2016, pp. 1–36. https://doi.org/10.1090/conm/669/13421
Bezuglyi, S., Kwiatkowski, J., and Medynets, K., Aperiodic substitution systems and their Bratteli diagrams, Ergodic Theory Dynam. Systems 29 (2009), no. 1, 37–72. https://doi.org/10.1017/S0143385708000230
Bezuglyi, S., Kwiatkowski, J., Medynets, K., and Solomyak, B., Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Systems 30 (2010), no. 4, 973–1007. https://doi.org/10.1017/S0143385709000443
Bezuglyi, S., Kwiatkowski, J., Medynets, K., and Solomyak, B., Finite rank Bratteli diagrams: structure of invariant measures, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2637–2679. https://doi.org/10.1090/S0002-9947-2012-05744-8
Dai, I., Garcia, X., Pădurariu, T., and Silva, C. E., On rationally ergodic and rationally weakly mixing rank-one transformations, Ergodic Theory Dynam. Systems 35 (2015), no. 4, 1141–1164. https://doi.org/10.1017/etds.2013.96
Danilenko, A. I., Strong orbit equivalence of locally compact Cantor minimal systems, Internat. J. Math. 12 (2001), no. 1, 113–123. https://doi.org/10.1142/S0129167X0100068X
Danilenko, A. I., Infinite rank one actions and nonsingular Chacon transformations, Illinois J. Math. 48 (2004), no. 3, 769–786.
Danilenko, A. I., Rank-one actions, their $(C,F)$-models and constructions with bounded parameters, eprint arxiv:1610.09851 [math.DS], 2016.
Downarowicz, T. and Maass, A., Finite-rank Bratteli-Vershik diagrams are expansive, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 739–747. https://doi.org/10.1017/S0143385707000673
Durand, F., Combinatorics on Bratteli diagrams and dynamical systems, in “Combinatorics, automata and number theory”, Encyclopedia Math. Appl., vol. 135, Cambridge Univ. Press, Cambridge, 2010, pp. 324–372.
Eigen, S., Hajian, A., Ito, Y., and Prasad, V., Weakly wandering sequences in ergodic theory, Springer Monographs in Mathematics, Springer, Tokyo, 2014. https://doi.org/10.1007/978-4-431-55108-9
El Abdalaoui, E. H., Lemańczyk, M., and de la Rue, T., On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal. 266 (2014), no. 1, 284–317. https://doi.org/10.1016/j.jfa.2013.09.005
Ferenczi, S., Systems of finite rank, Colloq. Math. 73 (1997), no. 1, 35–65. https://doi.org/10.4064/cm-73-1-35-65
Ferenczi, S. and Monteil, T., Infinite words with uniform frequencies, and invariant measures, in “Combinatorics, automata and number theory”, Encyclopedia Math. Appl., vol. 135, Cambridge Univ. Press, Cambridge, 2010, pp. 373–409.
Foreman, M. and Weiss, B., A characterization of odometer based systems: Working notes, preprint (personal communication), 2016.
Foreman, M. and Weiss, B., From odometers to circular systems: A global structure theorem, eprint arxiv:1703.07093 [math.DS], 2017.
Frick, S., Petersen, K., and Shields, S., Dynamical properties of some adic systems with arbitrary orderings, Ergodic Theory Dynam. Systems 37 (2017), no. 7, 2131–2162. https://doi.org/10.1017/etds.2015.128
Frick, S. B., Limited scope adic transformations, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), no. 2, 269–285. https://doi.org/10.3934/dcdss.2009.2.269
Herman, R. H., Putnam, I. F., and Skau, C. F., Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827–864. https://doi.org/10.1142/S0129167X92000382
Jewett, R. I., The prevalence of uniquely ergodic systems, J. Math. Mech. 19 (1969/1970), 717–729.
Kalikow, S. A., Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory Dynam. Systems 4 (1984), no. 2, 237–259. https://doi.org/10.1017/S014338570000242X
Krieger, W., On unique ergodicity, in “Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory”, 1972, pp. 327–346.
Lothaire, M., Applied combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 17, Addison-Wesley, Reading, Mass., 1983, a collective work by D. Perrin, et al.
Lothaire, M., Applied combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 105, Cambridge University Press, Cambridge, 2005, a collective work by J. Berstel, et al. https://doi.org/10.1017/CBO9781107341005
Medynets, K., Cantor aperiodic systems and Bratteli diagrams, C. R. Math. Acad. Sci. Paris 342 (2006), no. 1, 43–46. https://doi.org/10.1016/j.crma.2005.10.024
Méla, X., A class of nonstationary adic transformations, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 1, 103–123. https://doi.org/10.1016/j.anihpb.2005.02.002
Méla, X. and Petersen, K., Dynamical properties of the Pascal adic transformation, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 227–256. https://doi.org/10.1017/S0143385704000173
Mela, X. S., Dynamical properties of the Pascal adic and related systems, Ph.D. thesis, The University of North Carolina at Chapel Hill, 2002.
Yuasa, H., Invariant measures for the subshifts arising from non-primitive substitutions, J. Anal. Math. 102 (2007), 143–180. https://doi.org/10.1007/s11854-007-0019-8
Downloads
Published
2020-05-06
How to Cite
Frick, S., Petersen, K., & Shields, S. (2020). Periodic codings of Bratteli-Vershik systems. MATHEMATICA SCANDINAVICA, 126(2), 298–320. https://doi.org/10.7146/math.scand.a-117570
Issue
Section
Articles
