Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-19T13:11:44.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Pattern generation problems arising in multiplicative integer systems

Published online by Cambridge University Press:  25 September 2017

JUNG-CHAO BAN ,
WEN-GUEI HU  and
SONG-SUN LIN
JUNG-CHAO BAN
Affiliation:
Department of Applied Mathematics, National Dong Hwa University, Hualien 97401, Taiwan email jcban@gms.ndhu.edu.tw
WEN-GUEI HU
Affiliation:
College of Mathematics, Sichuan University, Chengdu, 610064, China email wghu@scu.edu.cn
SONG-SUN LIN
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, Hsin-Chu 30010, Taiwan email sslin@math.nctu.edu.tw
Get access Rights & Permissions [Opens in a new window]

Abstract

This study investigates a multiplicative integer system, an invariant subset of the full shift under the action of the semigroup of multiplicative integers, by using a method that was developed for studying pattern generation problems. The spatial entropy and the Minkowski dimensions of general multiplicative systems can thus be computed. A coupled system is the intersection of a multiplicative integer system and the golden mean shift, which can be decoupled by removing the multiplicative relation set and then performing procedures similar to those applied to a decoupled system. The spatial entropy can be obtained after the remaining error term is shown to approach zero.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1234 - 1260
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ban, J. C., Chang, C. H. and Lin, S. S.. On the structure of multi-layer cellular neural networks. J. Differential Equations 252(8) (2012), 4563–4597.Google Scholar
Ban, J. C., Chang, C. H., Lin, S. S. and Lin, Y. H.. Spatial complexity in multi-layer cellular neural networks. J. Differential Equations 246 (2009), 552–580.Google Scholar
Ban, J. C., Hu, W. G., Lin, S. S. and Lin, Y. H.. Verification of mixing properties in two-dimensional shifts of finite type. Preprint, 2011, arXiv:1112.2471.Google Scholar
Ban, J. C., Hu, W. G., Lin, S. S. and Lin, Y. H.. Zeta functions for two-dimensional shifts of finite type. Mem. Amer. Math. Soc. 221(1037) (2013).Google Scholar
Ban, J. C. and Lin, S. S.. Patterns generation and transition matrices in multi-dimensional lattice models. Discrete Contin. Dyn. Syst. 13(3) (2005), 637–658.Google Scholar
Ban, J. C., Lin, S. S. and Lin, Y. H.. Patterns generation and spatial entropy in two dimensional lattice models. Asian J. Math. 11(3) (2007), 497–534.Google Scholar
Ban, J. C., Lin, S. S. and Lin, Y. H.. Three-dimensional cellular neural networks and pattern generation problems. Intl J. Bifur. Chaos 18(4) (2008), 957–984.Google Scholar
Bowen, R.. Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.Google Scholar
Boyle, M., Pavlov, R. and Schraudner, M.. Multidimensional sofic shifts without separation and their factors. Trans. Amer. Math. Soc. 362 (2010), 4617–4653.Google Scholar
Burton, R. and Steif, J. E.. Non-uniqueness of measures of maximal entropy for subshifts of finite type. Ergod. Th. & Dynam. Sys. 14(2) (1994), 213–235.Google Scholar
Burton, R. and Steif, J. E.. New results on measures of maximal entropy. Israel J. Math. 89 (1995), 275–300.Google Scholar
Chow, S. N., Mallet-Paret, J. and Van Vleck, E. S.. Pattern formation and spatial chaos in spatially discrete evolution equations. Random Comput. Dynam. 4 (1996), 109–178.Google Scholar
Fan, A. H.. Some aspects of multifractal analysis. Geometry and Analysis of Fractals (Hong Kong, December 2012) (Springer Proceedings in Mathematics & Statistics, 88) . Springer, Berlin, 2014, pp. 115–145.Google Scholar
Fan, A. H., Liao, L. M. and Ma, J. H.. Level sets of multiple ergodic averages. Monatsh. Math. 168 (2012), 17–26.Google Scholar
Fan, A. H., Schmeling, J. and Wu, M.. Multifractal analysis of multiple ergodic averages. C. R. Math. Acad. Sci. Paris 349 (2011), 961–964.Google Scholar
Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1 (1967), 1–49.Google Scholar
Grünbaum, B. and Shephard, G. C.. Tilings and Patterns. W. H. Freeman, New York, 1986.Google Scholar
Häggström, O.. A subshift of finite type that is equivalent to the Ising model. Ergod. Th. & Dynam. Sys. 15 (1995), 543–556.Google Scholar
Haydn, N. T. A. and Ruelle, D.. Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification. Commun. Math. Phys. 148 (1992), 155–167.Google Scholar
Hochman, M. and Meyerovitch, T.. A characterization of the entropies of multidimensional shifts of finite type. Ann. Math. 171 (2010), 2011–2038.Google Scholar
Hu, W. G. and Lin, S. S.. Zeta functions for higher-dimensional shifts of finite type. Intl J. Bifur. Chaos 19(11) (2009), 3671–3689.Google Scholar
Hu, W. G. and Lin, S. S.. Nonemptiness problems of plane square tiling with two colors. Proc. Amer. Math. Soc. 139 (2011), 1045–1059.Google Scholar
Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161 (2005), 397–488.Google Scholar
Juang, J. and Lin, S. S.. Cellular neural networks: mosaic pattern and spatial chaos. SIAM J. Appl. Math. 60(3) (2000), 891–915.Google Scholar
Kenyon, R., Peres, Y. and Solomyak, B.. Hausdorff dimension of the multiplicative golden mean shift. C. R. Math. Acad. Sci. Paris 349 (2011), 625–628.Google Scholar
Kenyon, R., Peres, Y. and Solomyak, B.. Hausdorff dimension for fractals invariant under the multiplicative integers. Ergod. Th. & Dynam. Sys. 32 (2012), 1567–1584.Google Scholar
Lightwood, S. J.. Morphisms from non-periodic ℤ2 subshifts I: Constructing embeddings from homomorphisms. Ergod. Th. & Dynam. Sys. 23 (2003), 587–609.Google Scholar
Lightwood, S. J.. Morphisms from non-periodic ℤ2 subshifts II: Constructing homomorphisms to square-filling mixing shifts of finite type. Ergod. Th. & Dynam. Sys. 24 (2004), 1227–1260.Google Scholar
Lin, S. S. and Yang, T. S.. On the spatial entropy and patterns of two-dimensional cellular neural networks. Intl J. Bifur. Chaos 12 (2002), 115–128.Google Scholar
Lind, D.. A Zeta Function for ℤ d -Actions (London Mathematical Society Lecture Note Series, 228) . Cambridge University Press, Cambridge, 1996, pp. 433–450.Google Scholar
Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.Google Scholar
Markley, N. G. and Paul, M. E.. Maximal Measures and Entropy for ℤ𝜈 Subshifts of Finite Type. Classical Mechanics and Dynamical Systems (Lecture Notes in Pure and Appl. Math., 70) . Eds. Devaney, R. L. and Nitecki, Z. H.. Marcel Dekker, New York, 1979, pp. 135–157.Google Scholar
Peres, Y., Schmeling, J., Seuret, S. and Solomyak, B.. Dimensions of some fractals defined via the semigroup generated by 2 and 3. Israel J. Math. 199 (2014), 687–709.Google Scholar
Peres, Y. and Solomyak, B.. Dimension spectrum for a nonconventional ergodic average. Real Anal. Exchange 37 (2012), 375–388.Google Scholar
Pesin, Y.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications. University of Chicago Press, Chicago, 2008.Google Scholar
Pesin, Y. and Weiss, H.. The multifractal analysis of Birkhoff averages and large deviation. Global Analysis of Dynamical Systems. Institute of Physics, Bristol, 2001, pp. 419–431.Google Scholar
Quas, A. N. and Trow, P. B.. Subshifts of multidimensional shifts of finite type. Ergod. Th. & Dynam. Sys. 20 (2000), 859–874.Google Scholar
Ruelle, D.. Thermodynamic Formalism (Encyclopedia of Mathematics and its Applications, 5) . Addison-Wesley, Reading, MA, 1978.Google Scholar
Sloane, N. J. A. and Plouffe, S.. The Encyclopedia of Integer Sequences. Academic Press, New York, 1995.Google Scholar
Ward, T.. Automorphisms of ℤ d -subshifts of finite type. Indag. Math. (N.S.) 5(4) (1994), 495–504.Google Scholar
Ward, T.. Zeta functions for higher-dimensional actions. Preprint, University of East Anglia, March, 1994.Google Scholar