Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Arcwise connectedness of the set of ergodic measures of hereditary shifts

Authors: Jakub Konieczny, Michal Kupsa and Dominik Kwietniak
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 37B05; Secondary 37A35, 37B10, 37B40, 37D20
Published electronically: March 30, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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
  • [2] M. Avdeeva, Variance of $ \mathscr {B}$-free integers in short intervals, preprint, 2015, arXiv:1512.00149 [math.DS]
  • [3] 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
  • [4] John Banks, Piotr Oprocha, and Brett Stanley, Transitive sofic spacing shifts, Discrete Contin. Dyn. Syst. 35 (2015), no. 10, 4743-4764. MR 3392646
  • [5] 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
  • [6] 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
  • [7] Vaughn Climenhaga's Math Blog (accessed on September 26, 2016).
  • [8] 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
  • [9] Tomasz Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math. 74 (1991), no. 2-3, 241-256. MR 1135237
  • [10] Tomasz Downarowicz and Jacek Serafin, Possible entropy functions, Israel J. Math. 135 (2003), 221-250. MR 1997045
  • [11] Tomasz Downarowicz, Entropy in dynamical systems, New Mathematical Monographs, vol. 18, Cambridge University Press, Cambridge, 2011. MR 2809170
  • [12] 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
  • [13] Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1-49. MR 0213508
  • [14] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981. MR 603625
  • [15] K. Gelfert and D. Kwietniak, On density of ergodic measures and generic points, Ergodic Theory Dynam. Systems, FirstView, DOI: 10.1017/etds.2016.97
  • [16] Eli Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003. MR 1958753
  • [17] 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 0358312
  • [18] 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
  • [19] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, with a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. MR 1326374
  • [20] David Kerr and Hanfeng Li, Independence in topological and $ C^*$-dynamics, Math. Ann. 338 (2007), no. 4, 869-926. MR 2317754
  • [21] 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
  • [22] 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
  • [23] Kenneth Lau and Alan Zame, On weak mixing of cascades, Math. Systems Theory 6 (1972/73), 307-311. MR 0321058
  • [24] 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
  • [25] 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
  • [26] William Parry, Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge, 2004. Reprint of the 1981 original. MR 2140546
  • [27] Ryan Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow, Israel J. Math. 210 (2015), no. 1, 335-357. MR 3430278
  • [28] Anthony Quas and Ayşe A. Şahin, Entropy gaps and locally maximal entropy in $ \mathbb{Z}^d$ subshifts, Ergodic Theory Dynam. Systems 23 (2003), no. 4, 1227-1245. MR 1997974
  • [29] Anthony Quas and Terry Soo, Ergodic universality of some topological dynamical systems, Trans. Amer. Math. Soc. 368 (2016), no. 6, 4137-4170. MR 3453367
  • [30] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar 8 (1957), 477-493. MR 0097374
  • [31] 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
  • [32] Daniel J. Rudolph, Fundamentals of measurable dynamics, Ergodic theory on Lebesgue spaces, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. MR 1086631
  • [33] P. Sarnak, Three lectures on the Möbius function randomness and dynamics (Lecture 1),
  • [34] Paul C. Shields, The ergodic theory of discrete sample paths, Graduate Studies in Mathematics, vol. 13, American Mathematical Society, Providence, RI, 1996. MR 1400225
  • [35] M. Shinoda, Ergodic maximizing measures of non-generic, yet dense continuous functions, preprint, arXiv:1704.05616 [math.DS], 2017.
  • [36] S. M. Srivastava, A course on Borel sets, Graduate Texts in Mathematics, vol. 180, Springer-Verlag, New York, 1998. MR 1619545
  • [37] Brett Stanley, Bounded density shifts, Ergodic Theory Dynam. Systems 33 (2013), no. 6, 1891-1928. MR 3122156
  • [38] Peng Sun, Zero-entropy invariant measures for skew product diffeomorphisms, Ergodic Theory Dynam. Systems 30 (2010), no. 3, 923-930. MR 2643717
  • [39] Peng Sun, Measures of intermediate entropies for skew product diffeomorphisms, Discrete Contin. Dyn. Syst. 27 (2010), no. 3, 1219-1231. MR 2629583
  • [40] 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
  • [41] Benjamin Weiss, Single orbit dynamics, CBMS Regional Conference Series in Mathematics, vol. 95, American Mathematical Society, Providence, RI, 2000. MR 1727510
  • [42] Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37B05, 37A35, 37B10, 37B40, 37D20

Retrieve articles in all journals with MSC (2010): 37B05, 37A35, 37B10, 37B40, 37D20

Additional Information

Jakub Konieczny
Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG United Kingdom

Michal Kupsa
Affiliation: The Czech Academy of Sciences, Institute of Information Theory and Automation, Prague 8, CZ-18208 Czech Republic

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

Keywords: Kolmogorov-Sinai (metric) entropy, hereditary shift space, Poulsen simplex, Besicovitch pseudometric, $d$-bar metric
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
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society