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Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type
Faculty of Applied Mathematics, AGH University of Science and Technology, Poland, Krakow, Mickiewicza, A-3/A-4,305 |
In this text I study the asymptotics of the complexity function of minimal multidimensional subshifts of finite type through their entropy dimension, a topological invariant that has been introduced in order to study zero entropy dynamical systems. Following a recent trend in symbolic dynamics I approach this using concepts from computability theory. In particular it is known [
References:
[1] |
N. Aubrun and M. Sablik,
Multidimensional effective S-adic subshift are sofic, Unif. Distrib. Theory, 9 (2014), 7-29.
|
[2] |
A. Ballier, Propriétés Structurelles, Combinatoires et Logiques des Pavages, PhD Thesis, Aix-Marseille Université, 2009. |
[3] |
B. Durand and A. Romashchenko,
On the expressive power of quasiperiodic SFT, 42nd International Symposium on Mathematical Foundations of Computer Science, LIPIcs. Leibniz Int. Proc. Inform., Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 83 (2017), 1-14.
|
[4] |
B. Durand, L. A. Levin and A. Shen,
Complex tilings, J. Symbolic Logic, 73 (2008), 593-613.
doi: 10.2178/jsl/1208359062. |
[5] |
S. Gangloff and M. Sablik, Quantified block gluing, aperiodicity and entropy of multidimensional SFT, Journal d'Analyse Mathématique. |
[6] |
B. Grunbaum and G. C. Shepherd, Tilings and Patterns, W. H. Freeman and Company, New York, 1987. |
[7] |
M. Hochman and T. Meyerovitch,
A characterization of the entropies of multidimensional shifts of finite type, Ann. of Math., 171 (2010), 2011-2038.
doi: 10.4007/annals.2010.171.2011. |
[8] |
M. Hochman,
On the dynamics and recursive properties of multidimensional symbolic systems, Invent. Math., 176 (2009), 131-167.
doi: 10.1007/s00222-008-0161-7. |
[9] |
M. Hochman and P. Vanier,
Turing degree spectra of minimal subshifts, Computer Science - Theory and Applications, Lecture Notes in Comput. Springer, Cham, 10304 (2017), 154-161.
doi: 10.1007/978-3-319-58747-9_15. |
[10] |
U. Jung, J. Lee and K. Koh Park, Topological entropy dimension and directional entropy dimension for z2-subshifts, Entropy, 19 (2017), 13pp.
doi: 10.3390/e19020046. |
[11] |
E. Jeandel and P. Vanier,
Characterizations of periods of multidimensional shifts, Ergodic Theory Dynam. Systems, 35 (2015), 431-460.
doi: 10.1017/etds.2013.60. |
[12] |
T. Meyerovitch,
Growth-type invariants for zd subshifts of finite type and arithmetical classes of real numbers, Invent. Math., 184 (2011), 567-589.
doi: 10.1007/s00222-010-0296-1. |
[13] |
M. L. Minsky, Computation: Finite and infinite machines, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. |
[14] |
S. Mozes,
Tilings, substitution systems and dynamical systems generated by them, J. Analyse Math., 53 (1989), 139-186.
doi: 10.1007/BF02793412. |
[15] |
R. Pavlov and M. Schraudner,
Entropies realizable by block gluing shifts of finite type, J. Anal. Math., 126 (2015), 113-174.
doi: 10.1007/s11854-015-0014-4. |
[16] |
R. Robinson,
Undecidability and nonperiodicity for tilings of the plane, Invent. Math., 12 (1971), 177-209.
doi: 10.1007/BF01418780. |
show all references
References:
[1] |
N. Aubrun and M. Sablik,
Multidimensional effective S-adic subshift are sofic, Unif. Distrib. Theory, 9 (2014), 7-29.
|
[2] |
A. Ballier, Propriétés Structurelles, Combinatoires et Logiques des Pavages, PhD Thesis, Aix-Marseille Université, 2009. |
[3] |
B. Durand and A. Romashchenko,
On the expressive power of quasiperiodic SFT, 42nd International Symposium on Mathematical Foundations of Computer Science, LIPIcs. Leibniz Int. Proc. Inform., Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 83 (2017), 1-14.
|
[4] |
B. Durand, L. A. Levin and A. Shen,
Complex tilings, J. Symbolic Logic, 73 (2008), 593-613.
doi: 10.2178/jsl/1208359062. |
[5] |
S. Gangloff and M. Sablik, Quantified block gluing, aperiodicity and entropy of multidimensional SFT, Journal d'Analyse Mathématique. |
[6] |
B. Grunbaum and G. C. Shepherd, Tilings and Patterns, W. H. Freeman and Company, New York, 1987. |
[7] |
M. Hochman and T. Meyerovitch,
A characterization of the entropies of multidimensional shifts of finite type, Ann. of Math., 171 (2010), 2011-2038.
doi: 10.4007/annals.2010.171.2011. |
[8] |
M. Hochman,
On the dynamics and recursive properties of multidimensional symbolic systems, Invent. Math., 176 (2009), 131-167.
doi: 10.1007/s00222-008-0161-7. |
[9] |
M. Hochman and P. Vanier,
Turing degree spectra of minimal subshifts, Computer Science - Theory and Applications, Lecture Notes in Comput. Springer, Cham, 10304 (2017), 154-161.
doi: 10.1007/978-3-319-58747-9_15. |
[10] |
U. Jung, J. Lee and K. Koh Park, Topological entropy dimension and directional entropy dimension for z2-subshifts, Entropy, 19 (2017), 13pp.
doi: 10.3390/e19020046. |
[11] |
E. Jeandel and P. Vanier,
Characterizations of periods of multidimensional shifts, Ergodic Theory Dynam. Systems, 35 (2015), 431-460.
doi: 10.1017/etds.2013.60. |
[12] |
T. Meyerovitch,
Growth-type invariants for zd subshifts of finite type and arithmetical classes of real numbers, Invent. Math., 184 (2011), 567-589.
doi: 10.1007/s00222-010-0296-1. |
[13] |
M. L. Minsky, Computation: Finite and infinite machines, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. |
[14] |
S. Mozes,
Tilings, substitution systems and dynamical systems generated by them, J. Analyse Math., 53 (1989), 139-186.
doi: 10.1007/BF02793412. |
[15] |
R. Pavlov and M. Schraudner,
Entropies realizable by block gluing shifts of finite type, J. Anal. Math., 126 (2015), 113-174.
doi: 10.1007/s11854-015-0014-4. |
[16] |
R. Robinson,
Undecidability and nonperiodicity for tilings of the plane, Invent. Math., 12 (1971), 177-209.
doi: 10.1007/BF01418780. |
































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