# American Institute of Mathematical Sciences

February  2022, 42(2): 931-988. doi: 10.3934/dcds.2021143

## 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

Received  August 2019 Revised  April 2021 Published  February 2022 Early access  October 2021

Fund Project: The author is supported by NCN project no. 2019/35/B/ST1/02239

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 [12] that the possible values of entropy dimension for d-dimensional subshifts of finite type are the $\Delta_2$-computable numbers in $[0, d]$. The kind of constructions that underlies this result is however quite complex and minimality has been considered thus far as hard to achieve with it. In this text I prove that this is possible and use the construction principles that I developped in order to prove (in principle) that for all $d \ge 2$ the possible values for entropy dimensions of $d$-dimensional SFT are the $\Delta_2$-computable numbers in $[0, d-1]$. In the present text I prove formally this result for $d = 3$. Although the result for other dimensions does not follow directly, it is enough to understand this construction to see that it is possible to reproduce it in higher dimensions (I chose dimension three for optimality in terms of exposition). The case $d = 2$ requires some substantial changes to be made in order to adapt the construction that are not discussed here.

Citation: Silvère Gangloff. Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 931-988. doi: 10.3934/dcds.2021143
##### 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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##### 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.
The south west supertile of order two (left) and a representation of petals intersecting its support (right)
Correspondence between infinite supertiles and sub-patterns of a supertile of finite order
Schema of the proof. The separating line is colored gray
Illustration of the definition of the sets $\mathbb{O}_n$ and $\mathbb{C}_n$ (respectively dark gray set on the left and on the right)
Illustration of the construction of the functions $\varphi_n$
Decomposition of the pattern $p$ into a number $2^d$ of blocks
The set $\mathbb{O}_k$ can be decomposed into $2d$ cuboids. On this picture, $d = 2$
Illustration of the signal transformation in Meyerovitch's construction
Schema of the proof for the minimality property of $X_z$
Orientation of the faces of a three-dimensional cell. The symbols $h$ and $v$ indicate respectively the horizontal and vertical directions in each of the copies of the subshift $X_{\mathcal{R}}$
Simplified schema of the construction presented in this text. The cube represents a three-dimensional version of the cells observable in the Robinson subshift
Schema of the functional positions on the faces of a three-dimensional cell
Schema of the main rule for the orientation sublayer. The squares in the dashed areas are superimposed with the corresponding symbol. The large square at the center is the support petal immediately above in the hierarchy
Schematic illustration of the rule of the functional areas sublayer. Colors are used as a code for other illustrations
Pattern over the surface of a three-dimensional cell of order three in the functional areas sublayer. The arrows are oriented according to the fixed orientations of the face
Schemata of the transformation rules for the vertical addressing. On the two schemata on the left, the central petal has $p$-counter value not equal to $\overline{0}$. On the two schemata on the right, this value is $\overline{0}$
Illustration for the main rule of the horizontal addressing sublayer. The central petals on the two schemata on the left have $p$-counter value not equal to $\overline{0}$. The one on the schemata on the right have $p$-counter value equal to $\overline{0}$
Illustration for the active functional areas sublayer on a two-dimensional cell over the face of a three-dimensional cell, with $p = 3$. The addresses of active columns and and all the rows are represented
Notations for the faces of three-dimensional cells
Localization of the machine symbols on the bottom faces of the cubes, according to the direction ${\bf{e}}^3$. Blue columns (resp. rows) symbolize computation-active columns (resp. rows)
Schema of the inputs and outputs directions when inside the area (1) and on the border of the area (2, 3, 4, 5, 6)
Illustration of the standard rules (1)
Illustration of the standard rules (2)
Illustration of the propagation rule of the error signal, where are represented the empty tape, first machine and empty sides signals
Illustration of the transformation rules of the hierarchy bits when the grouping bit is $1$
Illustration of the completion of the $\texttt{on}/\texttt{off}$ signals and the space-time diagram of the machine. The known part is surrounded by a black square
Illustration of the completion of the arrows according to the error signal in the known part of the area, designated by a dashed rectangle
Schema of the proof for the minimality property of $X_z$
Illustration of the convolutions rules
Possible orientations of four neighbor supertiles having the same order (1)
Possible orientations of four neighbor supertiles having the same order (2)
Possible orientations of four neighbor supertiles having the same order (3)
Illustration of the correspondance between patterns of Figure 31 and parts of a supertile
Correspondence table for positions on the border of a face
Correspondence table for positions inside a face
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