Basic thermodynamic formalism for sandwich subshifts

Joanna Kułaga-Przymus; Michał D. Lemańczyk; Michał Rams

doi:10.3934/dcds.2025077

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2025, Volume 45, Issue 12: 4852-4901. Doi: 10.3934/dcds.2025077

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Basic thermodynamic formalism for sandwich subshifts

1.
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń, Chopina 12/18, 87-100 Toruń, Poland
2.
Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Toruń, Grudziądzka 5, 87-100 Toruń, Poland
3.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

^*Corresponding author: Joanna Kułaga-Przymus
^*Corresponding author: Joanna Kułaga-Przymus

Received: May 2024

Revised: April 2025

Early access: June 3, 2025

Published online: June 3, 2025

Research of the first author was supported by National Science Centre grant UMO-2019/33/B/ST1/00364 (Poland). Research of the last author was supported by National Science Centre grant 2019/33/B/ST1/00275 (Poland).

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Abstract

Define a partial order on $ \{0, 1\}^{\mathbb Z} $ such that $ x\leq y $ if and only if $ x_i\leq y_i $ for all $ i\in\mathbb{Z} $. A subshift $ X\subseteq\{0, 1\}^ {\mathbb{Z}} $ is hereditary if, for any $ x\in X $, it contains all $ y\in\{0, 1\}^\mathbb{Z} $ satisfying $ y\leq x $. Intuitively, a hereditary subshift contains all elements between its maximal elements (with respect to the aforementioned partial order) and $ 0^{\mathbb Z} $. A subshift is termed subordinate if it suffices to consider the orbit closure of all elements between a single maximal element $ x $ and $ 0^{\mathbb Z} $. This paper investigates measure-theoretic properties of such subshifts, focusing on thermodynamic formalism. The central concept is a measure-theoretic analogue of subordinate subshifts, wherein a single (maximal) invariant measure on $ \{0, 1\}^ {\mathbb{Z}} $ replaces the single maximal element.

Further, we introduce and analyze two-sided analogues of these classes: sandwich hereditary, sandwich subordinate, and sandwich measure-theoretically subordinate subshifts. Sandwich hereditary subshifts are defined as sets of elements bounded between pairs of maximal and minimal elements satisfying specified conditions. Sandwich subordinate subshifts arise when it suffices to consider the orbit closure of all elements between a single pair of sequences $ (w, x) $, where $ w \leq x $. Finally, in sandwich measure-theoretically subordinate subshifts, this pair of sequences is replaced by a pair of invariant measures on $ \{0, 1\}^ {\mathbb{Z}} $, specifically, by a joining of two such measures.

These notions and results are motivated by the theory of $ \mathscr{B} $-free systems.

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Mathematics Subject Classification: Primary: 37B10, 37A35; Secondary: 37D35, 28D20, 37A05.

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References

[1]	E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue, A dynamical point of view on the set of $\mathcal B$-free integers, Int. Math. Res. Not. IMRN, 2015 (2015), 7258-7286. doi: 10.1093/imrn/rnu164.
[2]	M. Baake and U. Grimm, Aperiodic Order. Vol. 1, vol. 149 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2013. A mathematical invitation, With a foreword by Roger Penrose.
[3]	V. Bergelson, T. Downarowicz and J. Vandehey, Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation, J. Anal. Math., 148 (2022), 213-286. doi: 10.1007/s11854-022-0226-3.
[4]	A. S. Besicovitch, On the density of certain sequences of integers, Math. Ann., 110 (1935), 336-341. doi: 10.1007/BF01448032.
[5]	E. Bessel-Hagen, Zahlentheorie, Teubner, 1929.
[6]	J.-R. Chazottes and G. Keller, Pressure and equilibrium states in ergodic theory, In Mathematics of Complexity and Dynamical Systems, Springer, New York, 1/3 (2012), 1422-1437. doi: 10.1007/978-1-4614-1806-1_90.
[7]	S. Chowla, On abundant numbers, J. Indian Math. Soc., New Ser., 1 (1934), 41-44.
[8]	H. Davenport, Über numeri abundantes, Preuss. Akad. Wiss. Sitzungsber, 26/29 (1933), 830-837.
[9]	H. Davenport and P. Erdös, On sequences of positive integers, Acta Arithmetica, 2 (1936), 147-151. doi: 10.4064/aa-2-1-147-151.
[10]	H. Davenport and P. Erdös, On sequences of positive integers, J. Indian Math. Soc. (N.S.), 15 (1951), 19-24.
[11]	T. Downarowicz, Survey of odometers and Toeplitz flows, In Algebraic and Topological Dynamics, Contemp. Math., Amer. Math. Soc., Providence, RI, 385 (2005), 7-37. doi: 10.1090/conm/385/07188.
[12]	T. Downarowicz, Entropy in Dynamical Systems, vol. 18 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2011.
[13]	A. Dymek, S. Kasjan and J. Kułaga-Przymus, Minimality of $\mathscr{B}$-free systems in number fields, Discrete Contin. Dyn. Syst., 43 (2023), 3512-3548. doi: 10.3934/dcds.2023056.
[14]	A. Dymek, S. Kasjan, J. Kułaga-Przymus and M. Lemańczyk, $\mathscr{B}$-free sets and dynamics, Trans. Amer. Math. Soc., 370 (2018), 5425-5489. doi: 10.1090/tran/7132.
[15]	A. Dymek, J. Kułaga-Przymus and D. Sell, Invariant measures for $\mathscr{B}$-free systems revisited, Ergodic Theory Dynam. Systems, 44 (2024), 2901-2932. doi: 10.1017/etds.2024.7.
[16]	P. Erdös, On the density of the abundant numbers, J. London Math. Soc., 9 (1934), 278-282. doi: 10.1112/jlms/s1-9.4.278.
[17]	R. M. Gray, Probability, Random Processes, and Ergodic Properties, second ed. Springer, Dordrecht, 2009.
[18]	R. R. Hall, Sets of Multiples, vol. 118 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.
[19]	R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech., 19 (1969/1970), 717-729. doi: 10.1512/iumj.1970.19.19058.
[20]	T. Kamae, Subsequences of normal sequences, Israel J. Math., 16 (1973), 121-149. doi: 10.1007/BF02757864.
[21]	S. Kasjan, G. Keller and M. Lemańczyk, Dynamics of $\mathcal{B}$-free sets: A view through the window, Int. Math. Res. Not. IMRN, 2019 (2019), 2690-2734. doi: 10.1093/imrn/rnx196.
[22]	S. Kasjan, M. Lemańczyk and S. Zuniga Alterman, Dynamics of $\mathscr{B}$-free systems generated by Behrend sets. Ⅰ, Acta Arith., 209 (2023), 135-171. doi: 10.4064/aa220525-14-2.
[23]	G. Keller, Generalized heredity in $\mathcal B$-free systems, Stoch. Dyn., 21 (2021), Paper No. 2140008, 19pp.
[24]	G. Keller, Tautness for sets of multiples and applications to $\mathcal B$-free dynamics, Studia Math., 247 (2019), 205-216. doi: 10.4064/sm180305-9-4.
[25]	G. Keller, Irregular $\mathcal{B}$-free Toeplitz sequences via Besicovitch's construction of sets of multiples without density, Monatsh. Math., 199 (2022), 801-816. doi: 10.1007/s00605-022-01754-6.
[26]	G. Keller and C. Richard, Dynamics on the graph of the torus parametrization, Ergodic Theory Dynam. Systems, 38 (2018), 1048-1085. doi: 10.1017/etds.2016.53.
[27]	J. Konieczny, M. Kupsa and D. Kwietniak, On $\overline{d}$-approachability, entropy density and $\mathscr{B}$-free shifts, Ergodic Theory Dynam. Systems, 43 (2023), 943-970. doi: 10.1017/etds.2021.167.
[28]	W. Krieger, 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, Univ. California Press, Berkeley, Calif., (1972), 327-346.
[29]	J. Kułaga-Przymus, M. Lemańczyk and B. Weiss, On invariant measures for $\mathscr{B}$-free systems, Proc. Lond. Math. Soc. (3), 110 (2015), 1435-1474. doi: 10.1112/plms/pdv017.
[30]	J. Kułaga-Przymus, M. Lemańczyk and B. Weiss, Hereditary subshifts whose simplex of invariant measures is Poulsen, In Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, Contemp. Math. Amer. Math. Soc., Providence, RI, 678 (2016), 245-253. doi: 10.1090/conm/678/13651.
[31]	J. Kułaga-Przymus and M. D. Lemańczyk, Hereditary subshifts whose measure of maximal entropy does not have the Gibbs property, Colloq. Math., 166 (2021), 107-127. doi: 10.4064/cm8223-11-2020.
[32]	J. Kułaga-Przymus and M. D. Lemańczyk, Entropy rate of product of independent processes, Monatsh. Math., 200 (2023), 131-162. doi: 10.1007/s00605-022-01801-2.
[33]	M. Kulczycki, D. Kwietniak and J. Li, Entropy of subordinate shift spaces, The American Mathematical Monthly, 125 (2018), 141-148. doi: 10.1080/00029890.2018.1401875.
[34]	D. Kwietniak, M. Łącka and P. Oprocha, Generic points for dynamical systems with average shadowing, Monatsh. Math., 183 (2017), 625-648. doi: 10.1007/s00605-016-1002-1.
[35]	M. D. Lemańczyk, Recurrence of stochastic processes in some concentration of measure and entropy problems, https://www.mimuw.edu.pl/sites/default/files/lemanczyk_rozprawa_doktorska.pdf.
[36]	Z. Lin and E. Chen, Equilibrium states which are not Gibbs measure on hereditary subshifts, https://arXiv.org/abs/2012.03409.
[37]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
[38]	B. Marcus and S. Newhouse, Measures of maximal entropy for a class of skew products, In Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach,) 1978, vol. 729 of Lecture Notes in Math., Springer, Berlin, (1979), 105-125.
[39]	L. Mirsky, Note on an asymptotic formula connected with $r$-free integers, Quart. J. Math. Oxford Ser., 18 (1947), 178-182. doi: 10.1093/qmath/os-18.1.178.
[40]	L. Mirsky, Arithmetical pattern problems relating to divisibility by $r$th powers, Proc. London Math. Soc. (2), 50 (1949), 497-508. doi: 10.1112/plms/s2-50.7.497.
[41]	R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow, Israel J. Math., 210 (2015), 335-357. doi: 10.1007/s11856-015-1255-8.
[42]	K. E. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1983.
[43]	F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, vol. 371 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2010.
[44]	P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sarnak/.
[45]	P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682.
[46]	P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Grad. Texts in Math., 79, Springer-Verlag, New York-Berlin, 1982.
[47]	B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math., 77 (1973), 462-474. doi: 10.1007/BF01295322.
[48]	S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete, 67 (1984), 95-107. doi: 10.1007/BF00534085.