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Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type



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

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  • 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.

    Mathematics Subject Classification: Primary: 37B50, 37B40; Secondary: 37B10, 68Q17.


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  • Figure 1.  The south west supertile of order two (left) and a representation of petals intersecting its support (right)

    Figure 2.  Correspondence between infinite supertiles and sub-patterns of a supertile of finite order

    Figure 3.  Schema of the proof. The separating line is colored gray

    Figure 4.  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)

    Figure 5.  Illustration of the construction of the functions $ \varphi_n $

    Figure 6.  Decomposition of the pattern $ p $ into a number $ 2^d $ of blocks

    Figure 7.  The set $ \mathbb{O}_k $ can be decomposed into $ 2d $ cuboids. On this picture, $ d = 2 $

    Figure 8.  Illustration of the signal transformation in Meyerovitch's construction

    Figure 9.  Schema of the proof for the minimality property of $ X_z $

    Figure 10.  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}} $

    Figure 11.  Simplified schema of the construction presented in this text. The cube represents a three-dimensional version of the cells observable in the Robinson subshift

    Figure 12.  Schema of the functional positions on the faces of a three-dimensional cell

    Figure 13.  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

    Figure 14.  Schematic illustration of the rule of the functional areas sublayer. Colors are used as a code for other illustrations

    Figure 15.  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

    Figure 16.  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} $

    Figure 17.  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} $

    Figure 18.  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

    Figure 19.  Notations for the faces of three-dimensional cells

    Figure 20.  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)

    Figure 21.  Schema of the inputs and outputs directions when inside the area (1) and on the border of the area (2, 3, 4, 5, 6)

    Figure 22.  Illustration of the standard rules (1)

    Figure 23.  Illustration of the standard rules (2)

    Figure 24.  Illustration of the propagation rule of the error signal, where are represented the empty tape, first machine and empty sides signals

    Figure 25.  Illustration of the transformation rules of the hierarchy bits when the grouping bit is $ 1 $

    Figure 26.  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

    Figure 27.  Illustration of the completion of the arrows according to the error signal in the known part of the area, designated by a dashed rectangle

    Figure 28.  Schema of the proof for the minimality property of $ X_z $

    Figure 29.  Illustration of the convolutions rules

    Figure 30.  Possible orientations of four neighbor supertiles having the same order (1)

    Figure 31.  Possible orientations of four neighbor supertiles having the same order (2)

    Figure 32.  Possible orientations of four neighbor supertiles having the same order (3)

    Figure 33.  Illustration of the correspondance between patterns of Figure 31 and parts of a supertile

    Table 1.  Correspondence table for positions on the border of a face

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    Table 2.  Correspondence table for positions inside a face

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