Arcwise connectedness of the set of ergodic measures of hereditary shifts
HTML articles powered by AMS MathViewer
- by Jakub Konieczny, Michal Kupsa and Dominik Kwietniak PDF
- Proc. Amer. Math. Soc. 146 (2018), 3425-3438 Request permission
Abstract:
We show that the set of ergodic invariant measures of a shift space with a safe symbol (this includes all hereditary shifts) is arcwise connected when endowed with the $d$-bar metric. As a consequence the set of ergodic measures of such a shift is also arcwise connected in the weak-star topology, and the entropy function over this set attains all values in the interval between zero and the topological entropy of the shift (inclusive). The latter result is motivated by a conjecture of A. Katok.References
- El Houcein El Abdalaoui, Mariusz Lemańczyk, and Thierry de la Rue, A dynamical point of view on the set of $\mathcal B$-free integers, Int. Math. Res. Not. IMRN 16 (2015), 7258–7286. MR 3428961, DOI 10.1093/imrn/rnu164
- M. Avdeeva, Variance of $\mathscr {B}$-free integers in short intervals, preprint, 2015, arXiv:1512.00149 [math.DS]
- John Banks, Thi T. D. Nguyen, Piotr Oprocha, Brett Stanley, and Belinda Trotta, Dynamics of spacing shifts, Discrete Contin. Dyn. Syst. 33 (2013), no. 9, 4207–4232. MR 3038059, DOI 10.3934/dcds.2013.33.4207
- John Banks, Piotr Oprocha, and Brett Stanley, Transitive sofic spacing shifts, Discrete Contin. Dyn. Syst. 35 (2015), no. 10, 4743–4764. MR 3392646, DOI 10.3934/dcds.2015.35.4743
- A. Bartnicka, S. Kasjan, J. Kułaga-Przymus, and M. Lemańczyk, $\mathscr {B}$-free sets and dynamics, Trans. Amer. Math. Soc., AMS Early View version, DOI https://doi.org/10.1090/tran/7132.
- F. Cellarosi and Ya. G. Sinai, Ergodic properties of square-free numbers, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1343–1374. MR 3055764, DOI 10.4171/JEMS/394
- Vaughn Climenhaga’s Math Blog (accessed on September 26, 2016). http://vaughnclimenhaga.wordpress.com/2012/04/18/a-useful-example-for-the-space- of-ergodic-measures/
- Vaughn Climenhaga and Daniel J. Thompson, Intrinsic ergodicity beyond specification: $\beta$-shifts, $S$-gap shifts, and their factors, Israel J. Math. 192 (2012), no. 2, 785–817. MR 3009742, DOI 10.1007/s11856-012-0052-x
- Tomasz Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math. 74 (1991), no. 2-3, 241–256. MR 1135237, DOI 10.1007/BF02775789
- Tomasz Downarowicz and Jacek Serafin, Possible entropy functions, Israel J. Math. 135 (2003), 221–250. MR 1997045, DOI 10.1007/BF02776059
- Tomasz Downarowicz, Entropy in dynamical systems, New Mathematical Monographs, vol. 18, Cambridge University Press, Cambridge, 2011. MR 2809170, DOI 10.1017/CBO9780511976155
- Manfred Einsiedler and Thomas Ward, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. MR 2723325, DOI 10.1007/978-0-85729-021-2
- Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 213508, DOI 10.1007/BF01692494
- H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
- K. Gelfert and D. Kwietniak, On density of ergodic measures and generic points, Ergodic Theory Dynam. Systems, FirstView, DOI: 10.1017/etds.2016.97
- Eli Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003. MR 1958753, DOI 10.1090/surv/101
- Richard Haydon, A new proof that every Polish space is the extreme boundary of a simplex, Bull. London Math. Soc. 7 (1975), 97–100. MR 358312, DOI 10.1112/blms/7.1.97
- Anatole Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 1245–1253. MR 804774
- Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
- David Kerr and Hanfeng Li, Independence in topological and $C^*$-dynamics, Math. Ann. 338 (2007), no. 4, 869–926. MR 2317754, DOI 10.1007/s00208-007-0097-z
- Joanna Kułaga-Przymus, Mariusz Lemańczyk, and Benjamin Weiss, On invariant measures for $\mathcal {B}$-free systems, Proc. Lond. Math. Soc. (3) 110 (2015), no. 6, 1435–1474. MR 3356811, DOI 10.1112/plms/pdv017
- Joanna Kułaga-Przymus, Mariusz Lemańczyk, and Benjamin Weiss, Hereditary subshifts whose simplex of invariant measures is Poulsen, Ergodic theory, dynamical systems, and the continuing influence of John C. Oxtoby, Contemp. Math., vol. 678, Amer. Math. Soc., Providence, RI, 2016, pp. 245–253. MR 3589826, DOI 10.1090/conm/678
- Kenneth Lau and Alan Zame, On weak mixing of cascades, Math. Systems Theory 6 (1972/73), 307–311. MR 321058, DOI 10.1007/BF01740722
- J. Lindenstrauss, G. Olsen, and Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, vi, 91–114 (English, with French summary). MR 500918
- Dominik Kwietniak, Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts, Discrete Contin. Dyn. Syst. 33 (2013), no. 6, 2451–2467. MR 3007694, DOI 10.3934/dcds.2013.33.2451
- William Parry, Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge, 2004. Reprint of the 1981 original. MR 2140546
- Ryan Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow, Israel J. Math. 210 (2015), no. 1, 335–357. MR 3430278, DOI 10.1007/s11856-015-1255-8
- Anthony Quas and Ayşe A. Şahin, Entropy gaps and locally maximal entropy in $\Bbb Z^d$ subshifts, Ergodic Theory Dynam. Systems 23 (2003), no. 4, 1227–1245. MR 1997974, DOI 10.1017/S0143385702001761
- Anthony Quas and Terry Soo, Ergodic universality of some topological dynamical systems, Trans. Amer. Math. Soc. 368 (2016), no. 6, 4137–4170. MR 3453367, DOI 10.1090/tran/6489
- A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 97374, DOI 10.1007/BF02020331
- E. Arthur Robinson Jr. and Ayşe A. Şahin, On the absence of invariant measures with locally maximal entropy for a class of $\mathbf Z^d$ shifts of finite type, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3309–3318. MR 1646203, DOI 10.1090/S0002-9939-99-05215-6
- Daniel J. Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. Ergodic theory on Lebesgue spaces. MR 1086631
- P. Sarnak, Three lectures on the Möbius function randomness and dynamics (Lecture 1), http://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf.
- Paul C. Shields, The ergodic theory of discrete sample paths, Graduate Studies in Mathematics, vol. 13, American Mathematical Society, Providence, RI, 1996. MR 1400225, DOI 10.1090/gsm/013
- M. Shinoda, Ergodic maximizing measures of non-generic, yet dense continuous functions, preprint, arXiv:1704.05616 [math.DS], 2017.
- S. M. Srivastava, A course on Borel sets, Graduate Texts in Mathematics, vol. 180, Springer-Verlag, New York, 1998. MR 1619545, DOI 10.1007/978-3-642-85473-6
- Brett Stanley, Bounded density shifts, Ergodic Theory Dynam. Systems 33 (2013), no. 6, 1891–1928. MR 3122156, DOI 10.1017/etds.2013.38
- Peng Sun, Zero-entropy invariant measures for skew product diffeomorphisms, Ergodic Theory Dynam. Systems 30 (2010), no. 3, 923–930. MR 2643717, DOI 10.1017/S0143385709000376
- Peng Sun, Measures of intermediate entropies for skew product diffeomorphisms, Discrete Contin. Dyn. Syst. 27 (2010), no. 3, 1219–1231. MR 2629583, DOI 10.3934/dcds.2010.27.1219
- Raúl Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part, Proc. Amer. Math. Soc. 140 (2012), no. 6, 1973–1985. MR 2888185, DOI 10.1090/S0002-9939-2011-11040-2
- Benjamin Weiss, Single orbit dynamics, CBMS Regional Conference Series in Mathematics, vol. 95, American Mathematical Society, Providence, RI, 2000. MR 1727510, DOI 10.1090/cbms/095
- Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581
Additional Information
- Jakub Konieczny
- Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG United Kingdom
- MR Author ID: 1178795
- Email: jakub.konieczny@gmail.com
- Michal Kupsa
- Affiliation: The Czech Academy of Sciences, Institute of Information Theory and Automation, Prague 8, CZ-18208 Czech Republic
- MR Author ID: 747570
- Email: kupsa@utia.cas.cz
- Dominik Kwietniak
- Affiliation: Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, ul. Łojasiewicza 6, 30-348 Kraków, Poland – and – Institute of Mathematics, Federal University of Rio de Janeiro, Cidade Universitaria - Ilha do Fundão, Rio de Janeiro 21945-909, Brazil
- MR Author ID: 773622
- Email: dominik.kwietniak@uj.edu.pl
- Received by editor(s): October 13, 2016
- Received by editor(s) in revised form: October 18, 2017, and November 19, 2017
- Published electronically: March 30, 2018
- Additional Notes: The first author was supported by the National Science Centre (NCN) under grant 2012/07/A/ST1/00185
The third author was supported by the National Science Centre (NCN) grant 2013/08/A/ST1/00275 and partially supported by CAPES/Brazil grant no. 88881.064927/2014-01. - Communicated by: Nimish Shah
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3425-3438
- MSC (2010): Primary 37B05; Secondary 37A35, 37B10, 37B40, 37D20
- DOI: https://doi.org/10.1090/proc/14029
- MathSciNet review: 3803667