Abstract
A multiplicative integer subshift \(X_\Omega \) derived from the subshift \(\Omega \) is invariant under multiplicative integer action, which is closely related to the level set of multiple ergodic average. The complexity of \(X_{\Omega }\) is usually measured by entropy (or box dimension). This work concerns on two types of multi-dimensional multiplicative integer subshifts (MMIS) with different coupling constraints, and then obtains their entropy formulae.
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Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114(2), 309–319 (1965)
Ban, J.C., Lin, S.S.: Patterns generation and transition matrices in multi-dimensional lattice models. Discret. Contin. Dyn. Syst. A 13(3), 637–658 (2005)
Ban, J.C., Lin, S.S., Lin, Y.H.: Patterns generation and spatial entropy in two dimensional lattice models. Asian J. Math. 10(3), 497–534 (2007)
Ban, J.C., Hu, W.G., Lin, S.S.: Pattern generation problems arising in multiplicative integer systems. Ergod. Theor. Dyn. Syst. 39, 1234–1260 (2019)
Barreira, L.: Dimension and Recurrence in Hyperbolic Dynamics, vol. 272. Springer, Berlin (2008)
Bourgain, J.: Double recurrence and almost sure convergence. J. Reine. Angew. Math. 404, 140–161 (1990)
Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153, 401–414 (1971)
Boyle, M., Pavlov, R., Schraudner, M.: Multidimensional sofic shifts without separation and their factors. Trans. Am. Math. Soc. 362, 4617–4653 (2010)
Carinci, G., Chazottes, J.R., Giardinà, C., Redig, F.: Nonconventional averages along arithmetic progressions and lattice spin systems. Indag. Math. 23(3), 589–602 (2012)
Chazottes, J.R., Redig, F.: Thermodynamic formalism and large deviations for multiplication-invariant potentials on lattice spin systems. Electron. J. Probab. 19, 1–19 (2019)
Chen, J.Y., Chen, Y.J., Hu, W.G., Lin, S.S.: Spatial chaos of Wang tiles with two symbols. J. Math. Phys. 57, 637–658 (2015)
Chow, S.N., Mallet-Paret, J., Van Vleck, E.S.: Pattern formation and spatial chaos in spatially discrete evolution equations. Random. Comput. Dyn. 4, 109–178 (1996)
Conze, J.P., Lesigne, E.: Théoremes ergodiques pour des mesures diagonales. Bull. Soc. Math. Fr. 112, 143–175 (1984)
Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York-London-Sydney (2003)
Fan, A.H.: Some aspects of multifractal analysis. Springer. Geom. Funct. Anal. 88, 115–145 (2014)
Fan, A.H., Liao, L., Ma, J.H.: Level sets of multiple ergodic averages. Monatsh. Math. 168, 17–26 (2012)
Fan, A.H., Liao, L., Wu, M.: Multifractal analysis of some multiple ergodic averages in linear Cookie–Cutter dynamical systems. Math. Z. 290(1–2), 63–81 (2018)
Fan, A.H., Schmeling, J., Wu, M.: Multifractal analysis of some multiple ergodic averages. Adv. Math. 295, 271–333 (2016)
Feng, D.J., Huang, W.: Variational principles for topological entropies of subsets. J. Funct. Anal. 263, 2228–2254 (2012)
Furstenberg, H., Katznelson, Y., Ornstein, D.: The ergodic theoretical proof of Szemerédi’s theorem. Bull. Am. Math. Soc. 7, 527–552 (1982)
Hochman, M., Meyerovitch, T.: A characterization of the entropies of multidimensional shifts of finite type. Ann. Math. 171, 2011–2038 (2010)
Host, B., Kra, B.: Nonconventional ergodic averages and nilmanifolds. Ann. Math. 161, 397–488 (2005)
Kenyon, R., Peres, Y., Solomyak, B.: Hausdorff dimension of the multiplicative golden mean shift. C. R. Acad. Sci. Paris. 349, 625–628 (2011)
Kenyon, R., Peres, Y., Solomyak, B.: Hausdorff dimension for fractals invariant under multiplicative integers. Ergod. Theor. Dyn. Syst. 32, 1567–1584 (2012)
Mandelbrot, B.: Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire. C. R. Acad. Sci. Paris. 278, 355–358 (1974)
Markley, N.G., Paul, M.E.: Maximal measures and entropy for \(Z^{\nu }\) subshift of finite type. Class. Mech. Dyn. Syst. 70, 135–157 (1979)
Markley, N.G., Paul, M.E.: Matrix subshifts for \(Z^{\nu }\) symbolic dynamics. Proc. Lond. Math. Soc. 43, 251–272 (1981)
Peres, Y., Schmeling, J., Seuret, S., Solomyak, B.: Dimensions of some fractals defined via the semigroup generated by 2 and 3. Isr. J. Math. 199(2), 687–709 (2014)
Pesin, Y.: Dimension Theory in Dynamical Systems: Contemporary Views and Application. University of Chicago Press, Chicago (1997)
Pesin, Y., Weiss, H.: The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 7, 89–106 (1997)
Pollicott, M.: A nonlinear transfer operator theorem. J. Stat. Phys. 166(3–4), 516–524 (2017)
Walters, P.: An Introduction to Ergodic Theory. Springer-Verlag, New York (1982)
Ward, T.: Automorphisms of \(\mathbb{Z}^{d}\)-subshifts of finite type. Indag. Math. 5(4), 495–504 (1994)
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The authors wish to express their gratitude to the editor and the anonymous referees for their careful reading and useful suggestions, which make significant improvements of this work.
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Communicated by Alessandro Giuliani.
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Ban is partially supported by the Ministry of Science and Technology, ROC (Contract MOST 107-2115-M-259 -001 -MY2 and 107-2115-M-390 -002 -MY2). Hu is partially supported by the National Natural Science Foundation of China (Grant 11601355)
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Ban, JC., Hu, WG. & Lai, GY. On the Entropy of Multidimensional Multiplicative Integer Subshifts. J Stat Phys 182, 31 (2021). https://doi.org/10.1007/s10955-021-02703-7
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DOI: https://doi.org/10.1007/s10955-021-02703-7