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Computational Complexity pp 965–999Cite as

Ergodic Theory of Cellular Automata

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Article Outline

Glossary

Definition of the Subject

Introduction

Invariant Measures for CA

Limit Measures and Other Asymptotics

Measurable Dynamics

Entropy

Future Directions and Open Problems

Acknowledgments

Bibliography

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Abbreviations

Configuration space and the shift :

Let \( { {\mathbb{M}} } \) be a finitely generated group or monoid (usually abelian). Typically, \( { {\mathbb{M}}=\mathbb{N}:=\{0,1,2,\ldots\} } \) or \( {\mathbb{M}}=\mathbb{Z}:=\{\ldots, -1,0,1,2,\ldots\} \), or \( { {\mathbb{M}}=\mathbb{N}^E } \), \( { \mathbb{Z} } \) D, or \( { \mathbb{Z}^D\times\mathbb{N}^E } \) for some \( { D,E\in\mathbb{N} } \). In some applications, \( { {\mathbb{M}} } \) could be nonabelian (although usually amenable), but to avoid notational complexity we will generally assume \( { {\mathbb{M}} } \) is abelian and additive, with operation ‘+’.

Let \( { {\mathcal{A}} } \) be a finite set of symbols (called an alphabet ). Let \( { \mathcal{A}^{\mathbb{M}} } \) denote the set of all functions \( { {\mathbf{a}}\colon{\mathbb{M}}{\longrightarrow}{\mathcal{A}} } \), which we regard as \( { {\mathbb{M}} } \)?indexed configurations of elements in \( { {\mathcal{A}} } \). We write such a configuration as \( { {\mathbf{a}}=[a_{\textsf{m}}]_{{\textsf{m}}\in{\mathbb{M}}} } \), where \( { a_{\textsf{m}}\in{\mathcal{A}} } \) for all \( { {\textsf{m}}\in{\mathbb{M}} } \), and refer to \( { \mathcal{A}^{\mathbb{M}} } \) as configuration space .

Treat \( { {\mathcal{A}} } \) as a discrete topological space; then \( { {\mathcal{A}} } \) is compact (because it is finite), so \( { \mathcal{A}^{\mathbb{M}} } \) is compact in the Tychonoff product topology. In fact, \( { \mathcal{A}^{\mathbb{M}} } \) is a  Cantor space : it is compact, perfect, totally disconnected, and metrizable. For example, if \( { {\mathbb{M}}=\mathbb{Z}^D } \), then the standard metric on \( { \mathcal{A}^{\mathbb{Z}^D} } \) is defined \( { d({\mathbf{a}},{\mathbf{b}})=2^{-\Delta({\mathbf{a}},{\mathbf{b}})} } \), where \( { \Delta({\mathbf{a}},{\mathbf{b}}):=\min{\{|{\textsf{z}}|; a_{\textsf{z}}\neq b_{\textsf{z}}\}} } \).

Any \( { {\textsf{v}}\in{\mathbb{M}} } \), determines a continuous shift map \( { \sigma^v\colon \mathcal{A}^{\mathbb{M}}{\longrightarrow}\mathcal{A}^{\mathbb{M}} } \) defined by \( { \sigma^v (\mathbf{a})_{\textsf{m}} = a_{{\textsf{m}}+{\textsf{v}}} } \) for all \( { {\mathbf{a}}\in\mathcal{A}^{\mathbb{M}} } \) and \( { {\textsf{m}}\in{\mathbb{M}} } \). The set \( { \{\sigma^v\}_{{\textsf{v}}\in{\mathbb{M}}} } \) is then a continuous \( { {\mathbb{M}} } \)?action on \( { \mathcal{A}^{\mathbb{M}} } \), which we denote simply by “σ”.

If \( { {\mathbf{a}}\in\mathcal{A}^{\mathbb{M}} } \) and \( { {\mathbb{U}}\subset{\mathbb{M}} } \), then we define \( { {\mathbf{a}}_{\mathbb{U}}\in{\mathcal{A}}^{\mathbb{U}} } \) by \( { {\mathbf{a}}_{\mathbb{U}}:=[a_{\textsf{u}}]_{{\textsf{u}}\in{\mathbb{U}}} } \). If \( { {\textsf{m}}\in{\mathbb{M}} } \), then strictly speaking, \( { {\mathbf{a}}_{{\textsf{m}}+{\mathbb{U}}}\in{\mathcal{A}}^{{\textsf{m}}+{\mathbb{U}}} } \); however, it will often be convenient to ‘abuse notation’ and treat \( { {\mathbf{a}}_{{\textsf{m}}+{\mathbb{U}}} } \) as an element of \( { {\mathcal{A}}^{{\mathbb{U}}} } \) in the obvious way.

Cellular automata :

Let \( { \mathbb{H}\subset{\mathbb{M}} } \) be some finite subset, and let \( { \phi\colon{\mathcal{A}}^{\mathbb{H}}{\longrightarrow}{\mathcal{A}} } \) be a function (called a  local rule ). The cellular automaton (CA) determined by ? is the function \( { \Phi\colon\mathcal{A}^{\mathbb{M}}{\longrightarrow}\mathcal{A}^{\mathbb{M}} } \) defined by \( { \Phi({\mathbf{a}})_{{\textsf{m}}} = \phi({\mathbf{a}}_{{\textsf{m}}+\mathbb{H}}) } \) for all \( { {\mathbf{a}}\in\mathcal{A}^{\mathbb{M}} } \) and \( { {\textsf{m}}\in{\mathbb{M}} } \). Curtis, Hedlund and Lyndon showed that cellular automata are exactly the continuous transformations of \( { \mathcal{A}^{\mathbb{M}} } \) which commute with all shifts (see Theorem 3.4 in [58]). We refer to \( { \mathbb{H} } \) as the neighborhood of Φ. For example, if \( { {\mathbb{M}}=\mathbb{Z} } \), then typically \( { \mathbb{H}:=[-\ell \ldots r]:=\{-\ell,1-\ell,\ldots,r-1,r\} } \) for some left radius \( { \ell \geq 0 } \) and right radius \( { r \geq 0 } \). If \( { -\ell \geq 0 } \), then ? can either define CA on \( { \mathcal{A}^{\mathbb{N}} } \) or define a  one-sided CA on \( { \mathcal{A}^{\mathbb{Z}} } \). If \( { {\mathbb{M}}=\mathbb{Z}^D } \), then typically \( { \mathbb{H}\subseteq[-R \ldots R]^D } \), for some radius \( { R\geq 0 } \). Normally we assume that \( { \ell } \), r, and R are chosen to be minimal. Several specific classes of CA will be important to us:

Linear CA :

Let \( { ({\mathcal{A}},+) } \) be a finite abelian group (e.?g. \( { {\mathcal{A}}=\mathbb{Z}_{/p} } \), where \( { p\in\mathbb{N} } \); usually p is prime). Then Φ is a  linear CA (LCA) if the local rule ? has the form

$$ \phi({\mathbf{a}}_\mathbb{H}) := \sum_{\textsf{h}\in\mathbb{H}} \varphi_{\textsf{h}}(a_{\textsf{h}})\:,\quad \forall \mathbf{a}_\mathbb{H} \in \mathcal{A}^{\mathbb{H}}\:, $$
(1)

where \( { \varphi_{\textsf{h}}\colon\mathcal{A}{\longrightarrow}\mathcal{A} } \) is an endomorphism of \( { (\mathcal{A},+) } \), for each \( { \textsf{h}\in\mathbb{H} } \). We say that Φ has scalar coefficients if, for each \( { \textsf{h}\in\mathbb{H} } \), there is some scalar \( { c_{\textsf{h}}\in\mathbb{Z} } \), so that \( { \varphi_{\textsf{h}}(a_{\textsf{h}}) := c_{\textsf{h}} \cdot a_{\textsf{h}} } \); then \( { \phi({\mathbf{a}}_\mathbb{H}) := \sum_{\textsf{h}\in\mathbb{H}} c_{\textsf{h}} a_{\textsf{h}} } \). For example, if \( \mathcal{A}=(\mathbb{Z}_{/p},+) \), then all endomorphisms are scalar multiplications, so all LCA have scalar coefficients.

If \( { c_{\textsf{h}}=1 } \) for all \( { \textsf{h}\in\mathbb{H} } \), then Φ has local rule \( { \phi(\mathbf{a}_\mathbb{H}) := \sum_{\textsf{h}\in\mathbb{H}} a_{\textsf{h}} } \); in this case, Φ is called an additive cellular automaton ; see Additive Cellular Automata.

Affine CA :

If \( { ({\mathcal{A}},+) } \) is a finite abelian group, then an affine CA is one with a local rule \( \phi({\mathbf{a}}_\mathbb{H}) := c+ \sum_{\textsf{h}\in\mathbb{H}} \varphi_{\textsf{h}}(a_{\textsf{h}}) \), where c is some constant and where \( { \varphi_{\textsf{h}}\colon{\mathcal{A}}{\longrightarrow}{\mathcal{A}} } \) are endomorphisms of \( { ({\mathcal{A}},+) } \). Thus, Φ is an LCA if \( { c=0 } \).

Permutative CA :

Suppose \( \Phi\colon\mathcal{A}^{\mathbb{Z}}{\longrightarrow}\mathcal{A}^{\mathbb{Z}} \) has local rule \( \phi\colon{\mathcal{ A}}^{[-\ell \ldots r]}{\longrightarrow}{\mathcal{A}} \). Fix \( {\mathbf{b}}=[b_{1-\ell},\ldots, b_{r-1}, b_r]\in{\mathcal{A}}^{(-\ell \ldots r]} \). For any \( { a\in{\mathcal{A}} } \), define \( [a\;{\mathbf{b}}]:=[a,b_{1-\ell},\ldots,b_{r-1},b_r]\in {\mathcal{A}}^{[-\ell \ldots r]} \). We then define the function \( \phi_{\mathbf{b}}\colon{\mathcal{A}}{\longrightarrow}{\mathcal{A}} \) by \( \phi_{\mathbf{b}}(a) := \phi([a\;{\mathbf{b}}]) \). We say that Φ is left‐permutative if \( \phi_{\mathbf{b}}\colon{\mathcal{A}}{\longrightarrow}{\mathcal{A}} \) is a permutation (i.?e. a bijection) for all \( { \mathbf{b}\in\mathcal{A}^{(-\ell \ldots r]} } \).

Likewise, given \( { {\mathbf{b}}=[b_{-\ell},\ldots,b_{r-1}]\in{\mathcal{A}}^{[-\ell \ldots r)} } \) and \( { c\in{\mathcal{A}} } \), define \( [{\mathbf{b}}\;c]:=[b_{-\ell},b_{1-\ell}\ldots,b_{r-1},c]\in{\mathcal{A}}^{[-\ell \ldots r]} \), and define \( _{\mathbf{b}}\phi\colon{\mathcal{A}}{\longrightarrow}{\mathcal{A}} \) by \( _{\mathbf{b}}\phi(c) := \phi([{\mathbf{b}} c]) \); then Φ is right‐permutative if \( _{\mathbf{b}}\phi\colon {\mathcal{A}}{\longrightarrow} {\mathcal{A}} \) is a permutation for all \( {\mathbf{b}}\in{\mathcal{A}}^{[-\ell \ldots r)} \). We say Φ is bipermutative if it is both left- and right‐permutative. More generally, if \( { {\mathbb{M}} } \) is any monoid, \( { \mathbb{H}\subset{\mathbb{M}} } \) is any neighborhood, and \( { \textsf{h}\in\mathbb{H} } \) is any fixed coordinate, then we define h?permutativity for a CA on \( { \mathcal{A}^{\mathbb{M}} } \) in the obvious fashion.

For example, suppose \( { ({\mathcal{A}},+) } \) is an abelian group and Φ is an affine CA on \( { \mathcal{A}^{\mathbb{Z}} } \) with local rule \( \phi({\mathbf{a}}_\mathbb{H}) = c + \sum_{h=-\ell}^r \varphi_h(a_h) \). Then Φ is left‐permutative iff \( { \varphi_{-\ell} } \) is an automorphism, and right‐permutative iff \( { \varphi_r } \) is an automorphism. If \( { {\mathcal{A}}=\mathbb{Z}_{/p} } \), and p is prime, then every nontrivial endomorphism is an automorphism(because it is multiplication by a nonzero element of \( { \mathbb{Z}_{/p} } \), which is a field), so in this case, every affine CA is permutative in every coordinate of its neighborhood (and in particular, bipermutative). If \( { {\mathcal{A}}\neq\mathbb{Z}_{/p} } \), however, then not all affine CA are permutative.

Permutative CA were introduced by Hedlund [58], §6, and are sometimes called permutive CA. Right permutative CA on \( \smash{ \mathcal{A}^{\mathbb{N}} } \) are also called toggle automata . For more information, see Sect. 7 of Topological Dynamics of Cellular Automata.

Linear CA :

Let \( { ({\mathcal{A}},+) } \) be a finite abelian group (e.?g. \( { {\mathcal{A}}=\mathbb{Z}_{/p} } \), where \( { p\in\mathbb{N} } \); usually p is prime). Then Φ is a  linear CA (LCA) if the local rule ? has the form

$$ \phi({\mathbf{a}}_\mathbb{H}) := \sum_{\textsf{h}\in\mathbb{H}} \varphi_{\textsf{h}}(a_{\textsf{h}})\:,\quad \forall \mathbf{a}_\mathbb{H} \in \mathcal{A}^{\mathbb{H}}\:, $$
(1)

where \( { \varphi_{\textsf{h}}\colon\mathcal{A}{\longrightarrow}\mathcal{A} } \) is an endomorphism of \( { (\mathcal{A},+) } \), for each \( { \textsf{h}\in\mathbb{H} } \). We say that Φ has scalar coefficients if, for each \( { \textsf{h}\in\mathbb{H} } \), there is some scalar \( { c_{\textsf{h}}\in\mathbb{Z} } \), so that \( { \varphi_{\textsf{h}}(a_{\textsf{h}}) := c_{\textsf{h}} \cdot a_{\textsf{h}} } \); then \( { \phi({\mathbf{a}}_\mathbb{H}) := \sum_{\textsf{h}\in\mathbb{H}} c_{\textsf{h}} a_{\textsf{h}} } \). For example, if \( \mathcal{A}=(\mathbb{Z}_{/p},+) \), then all endomorphisms are scalar multiplications, so all LCA have scalar coefficients.

If \( { c_{\textsf{h}}=1 } \) for all \( { \textsf{h}\in\mathbb{H} } \), then Φ has local rule \( { \phi(\mathbf{a}_\mathbb{H}) := \sum_{\textsf{h}\in\mathbb{H}} a_{\textsf{h}} } \); in this case, Φ is called an additive cellular automaton ; see Additive Cellular Automata.

Affine CA :

If \( { ({\mathcal{A}},+) } \) is a finite abelian group, then an affine CA is one with a local rule \( \phi({\mathbf{a}}_\mathbb{H}) := c+ \sum_{\textsf{h}\in\mathbb{H}} \varphi_{\textsf{h}}(a_{\textsf{h}}) \), where c is some constant and where \( { \varphi_{\textsf{h}}\colon{\mathcal{A}}{\longrightarrow}{\mathcal{A}} } \) are endomorphisms of \( { ({\mathcal{A}},+) } \). Thus, Φ is an LCA if \( { c=0 } \).

Permutative CA :

Suppose \( \Phi\colon\mathcal{A}^{\mathbb{Z}}{\longrightarrow}\mathcal{A}^{\mathbb{Z}} \) has local rule \( \phi\colon{\mathcal{ A}}^{[-\ell \ldots r]}{\longrightarrow}{\mathcal{A}} \). Fix \( {\mathbf{b}}=[b_{1-\ell},\ldots, b_{r-1}, b_r]\in{\mathcal{A}}^{(-\ell \ldots r]} \). For any \( { a\in{\mathcal{A}} } \), define \( [a\;{\mathbf{b}}]:=[a,b_{1-\ell},\ldots,b_{r-1},b_r]\in {\mathcal{A}}^{[-\ell \ldots r]} \). We then define the function \( \phi_{\mathbf{b}}\colon{\mathcal{A}}{\longrightarrow}{\mathcal{A}} \) by \( \phi_{\mathbf{b}}(a) := \phi([a\;{\mathbf{b}}]) \). We say that Φ is left‐permutative if \( \phi_{\mathbf{b}}\colon{\mathcal{A}}{\longrightarrow}{\mathcal{A}} \) is a permutation (i.?e. a bijection) for all \( { \mathbf{b}\in\mathcal{A}^{(-\ell \ldots r]} } \).

Likewise, given \( { {\mathbf{b}}=[b_{-\ell},\ldots,b_{r-1}]\in{\mathcal{A}}^{[-\ell \ldots r)} } \) and \( { c\in{\mathcal{A}} } \), define \( [{\mathbf{b}}\;c]:=[b_{-\ell},b_{1-\ell}\ldots,b_{r-1},c]\in{\mathcal{A}}^{[-\ell \ldots r]} \), and define \( _{\mathbf{b}}\phi\colon{\mathcal{A}}{\longrightarrow}{\mathcal{A}} \) by \( _{\mathbf{b}}\phi(c) := \phi([{\mathbf{b}} c]) \); then Φ is right‐permutative if \( _{\mathbf{b}}\phi\colon {\mathcal{A}}{\longrightarrow} {\mathcal{A}} \) is a permutation for all \( {\mathbf{b}}\in{\mathcal{A}}^{[-\ell \ldots r)} \). We say Φ is bipermutative if it is both left- and right‐permutative. More generally, if \( { {\mathbb{M}} } \) is any monoid, \( { \mathbb{H}\subset{\mathbb{M}} } \) is any neighborhood, and \( { \textsf{h}\in\mathbb{H} } \) is any fixed coordinate, then we define h?permutativity for a CA on \( { \mathcal{A}^{\mathbb{M}} } \) in the obvious fashion.

For example, suppose \( { ({\mathcal{A}},+) } \) is an abelian group and Φ is an affine CA on \( { \mathcal{A}^{\mathbb{Z}} } \) with local rule \( \phi({\mathbf{a}}_\mathbb{H}) = c + \sum_{h=-\ell}^r \varphi_h(a_h) \). Then Φ is left‐permutative iff \( { \varphi_{-\ell} } \) is an automorphism, and right‐permutative iff \( { \varphi_r } \) is an automorphism. If \( { {\mathcal{A}}=\mathbb{Z}_{/p} } \), and p is prime, then every nontrivial endomorphism is an automorphism(because it is multiplication by a nonzero element of \( { \mathbb{Z}_{/p} } \), which is a field), so in this case, every affine CA is permutative in every coordinate of its neighborhood (and in particular, bipermutative). If \( { {\mathcal{A}}\neq\mathbb{Z}_{/p} } \), however, then not all affine CA are permutative.

Permutative CA were introduced by Hedlund [58], §6, and are sometimes called permutive CA. Right permutative CA on \( \smash{ \mathcal{A}^{\mathbb{N}} } \) are also called toggle automata . For more information, see Sect. 7 of Topological Dynamics of Cellular Automata.

Subshifts :

A subshift is a closed, σ?invariant subset \( \mathbf{X}\subset\mathcal{A}^{\mathbb{M}} \). For any \( \smash{ \mathbb{U}\subset{\mathbb{M}} } \), let \( \smash{\mathbf{X}}_{\mathbb{U}} := {\{{\mathbf{x}}_{\mathbb{U}}; {\mathbf{x}}\in{\mathbf{X}}\} }\subset {\mathcal{A}}^{\mathbb{U}} \). We say \( \smash{ {\mathbf{X}} } \) is a  subshift of finite type (SFT) if there is some finite \( \smash{ {\mathbb{U}}\subset{\mathbb{M}} } \) such that \( \smash{ {\mathbf{X}} } \) is entirely described by \( \smash{ {\mathbf{X}}_{\mathbb{U}} } \), in the sense that \( \smash{\mathbf{X}}={\{{\mathbf{x}}\in\mathcal{A}^{\mathbb{M}} ; {\mathbf{x}}_{{\mathbb{U}}+{\textsf{ m}}}\in{\mathbf{X}}_{\mathbb{U}}, \forall{\textsf{m}}\in{\mathbb{M}}\}} \).

In particular, if \( \smash{ {\mathbb{M}}=\mathbb{Z} } \), then a (two-sided) Markov subshift is an SFT \( \smash{ {\mathbf{X}}\subset\mathcal{A}^{\mathbb{Z}} } \) determined by a set \( \smash{ {\mathbf{X}}_{\{0,1\}}\subset{\mathcal{ A}}^{\{0,1\}} } \) of admissible transitions ; equivalently, \( \smash{ {\mathbf{X}} } \) is the set of all bi-infinite directed paths in a digraph whose vertices are the elements of \( \smash{ {\mathcal{A}} } \), with an edge \( \smash{ a\leadsto b } \) iff \( \smash{ (a,b)\in{\mathbf{X}}_{\{0,1\}} } \). If \( \smash{ {\mathbb{M}}=\mathbb{N} } \), then a  one-sided Markov subshift is a subshift of \( \smash{ \mathcal{A}^{\mathbb{N}} } \) defined in the same way.

If \( \smash{ D\geq 2 } \), then an SFT in \( \smash{ \mathcal{A}^{\mathbb{Z}^D} } \) can be thought of as the set of admissible ‘tilings’ of \( \smash{ \mathbb{R} } \) D by Wang tiles corresponding to the elements of \( \smash{ {\mathbf{X}}_{\mathbb{U}} } \). (Wang tiles are unit squares (or (hyper)cubes) with various ‘notches’ cut into their edges (or (hyper)faces) so that they can only be juxtaposed in certain ways.)

A subshift \( \smash{ {\mathbf{X}}\subseteq\mathcal{A}^{\mathbb{Z}^D} } \) is strongly irreducible (or topologically mixing ) if there is some \( \smash{ R\in\mathbb{N} } \) such that, for any disjoint finite subsets \( \smash{ {\mathbb{V}},{\mathbb{U}}\subset\mathbb{Z}^D } \) separated by a distance of at least R, and for any \( \smash{ {\mathbf{u}}\in{\mathbf{X}}_{\mathbb{U}} } \) and \( \smash{ {\mathbf{v}}\in{\mathbf{ X}}_{\mathbb{V}} } \), there is some \( \smash{ {\mathbf{x}}\in{\mathbf{X}} } \) such that \( \smash{ {\mathbf{ x}}_{\mathbb{U}}={\mathbf{u}} } \) and \( \smash{ {\mathbf{x}}_{\mathbb{V}}={\mathbf{v}} } \).

Measures :

For any finite subset \( \smash{ {\mathbb{U}}\subset{\mathbb{M}} } \), and any \( \smash{ {\mathbf{b}}\in{\mathcal{A}}^{\mathbb{U}} } \), let \( \smash{ \langle{\mathbf{b}}\rangle := {\{{\mathbf{a}}\in\mathcal{A}^{\mathbb{M}} ; {\mathbf{a}}_{\mathbb{U}}:={\mathbf{b}}\} } } \) be the cylinder set determined by \( \smash{ {\mathbf{b}} } \). Let \( \smash{ {\mathfrak{B}} } \) be the sigma-algebra on \( \smash{ \mathcal{A}^{\mathbb{M}} } \) generated by all cylinder sets. A (probability) measure μ on \( \smash{ \mathcal{A}^{\mathbb{M}} } \) is a countably additive function \( \smash{ \mu\colon{\mathfrak{ B}}{\longrightarrow}[0,1] } \) such that \( \smash{ \mu[\mathcal{A}^{\mathbb{M}}]=1 } \). A measure on \( \smash{ \mathcal{A}^{\mathbb{M}} } \) is entirely determined by its values on cylinder sets. We will be mainly concerned with the following classes of measures:

Bernoulli measure :

Let \( { \beta_0 } \) be a probability measure on \( { {\mathcal{A}} } \). The Bernoulli measure induced by \( { \beta_0 } \) is the measure β on \( { \mathcal{A}^{\mathbb{M}} } \) such that, for any finite subset \( { {\mathbb{U}}\subset{\mathbb{M}} } \), and any \( { {\mathbf{a}}\in{\mathcal{A}}^{\mathbb{U}} } \), if \( { U:=|{\mathbb{U}}| } \), then \( { \beta[\langle{\mathbf{a}}\rangle] = \prod_{\textsf{h}\in\mathbb{H}} \beta_0(a_{\textsf{h}}) } \).

Invariant measure :

Let μ be a measure on \( { \mathcal{A}^{\mathbb{M}} } \), and let \( { \Phi\colon\mathcal{A}^{\mathbb{M}}{\longrightarrow}\mathcal{A}^{\mathbb{M}} } \) be a cellular automaton. The measure \( { \Phi\mu } \) is defined by \( { \Phi\mu ({\mathbf{B}}) = \mu(\Phi^{-1}({\mathbf{B}})) } \), for any \( { {\mathbf{B}}\in{\mathfrak{B}} } \). We say that μ is Φ?invariant (or that Φ is μ?preserving ) if \( { \Phi\mu = \mu } \).

Uniform measure :

Let \( { A:=|{\mathcal{A}}| } \). The uniform measure η on \( { \mathcal{A}^{\mathbb{M}} } \) is the Bernoulli measure such that, for any finite subset \( { {\mathbb{U}}\subset{\mathbb{M}} } \), and any \( { {\mathbf{b}}\in{\mathcal{A}}^{\mathbb{U}} } \), if \( { U:=|{\mathbb{U}}| } \), then \( { \mu[\langle{\mathbf{b}}\rangle] = 1/A^U } \).

The support of a measure μ is the smallest closed subset \( { \mathbf{X}\subset\mathcal{A}^{\mathbb{M}} } \) such that \( { \mu[{\mathbf{X}}]=1 } \); we denote this by supp(μ). We say μ has full support if \( \textsf{supp}(\mu)=\mathcal{A}^{\mathbb{M}} \) – equivalently, \( { \mu[{\mathbf{C}}] > 0 } \) for every cylinder subset \( { {\mathbf{C}}\subset\mathcal{A}^{\mathbb{M}} } \).

Bernoulli measure :

Let \( { \beta_0 } \) be a probability measure on \( { {\mathcal{A}} } \). The Bernoulli measure induced by \( { \beta_0 } \) is the measure β on \( { \mathcal{A}^{\mathbb{M}} } \) such that, for any finite subset \( { {\mathbb{U}}\subset{\mathbb{M}} } \), and any \( { {\mathbf{a}}\in{\mathcal{A}}^{\mathbb{U}} } \), if \( { U:=|{\mathbb{U}}| } \), then \( { \beta[\langle{\mathbf{a}}\rangle] = \prod_{\textsf{h}\in\mathbb{H}} \beta_0(a_{\textsf{h}}) } \).

Invariant measure :

Let μ be a measure on \( { \mathcal{A}^{\mathbb{M}} } \), and let \( { \Phi\colon\mathcal{A}^{\mathbb{M}}{\longrightarrow}\mathcal{A}^{\mathbb{M}} } \) be a cellular automaton. The measure \( { \Phi\mu } \) is defined by \( { \Phi\mu ({\mathbf{B}}) = \mu(\Phi^{-1}({\mathbf{B}})) } \), for any \( { {\mathbf{B}}\in{\mathfrak{B}} } \). We say that μ is Φ?invariant (or that Φ is μ?preserving ) if \( { \Phi\mu = \mu } \).

Uniform measure :

Let \( { A:=|{\mathcal{A}}| } \). The uniform measure η on \( { \mathcal{A}^{\mathbb{M}} } \) is the Bernoulli measure such that, for any finite subset \( { {\mathbb{U}}\subset{\mathbb{M}} } \), and any \( { {\mathbf{b}}\in{\mathcal{A}}^{\mathbb{U}} } \), if \( { U:=|{\mathbb{U}}| } \), then \( { \mu[\langle{\mathbf{b}}\rangle] = 1/A^U } \).

Notation :

Let \( { {\textsf{C\!A}(\mathcal{A}^{\mathbb{M}})} } \) denote the set of all cellular automata on \( { \mathcal{A}^{\mathbb{M}} } \). If \( { {\mathbf{X}}\subset\mathcal{A}^{\mathbb{M}} } \), then let \( { {\textsf{C\!A}(\mathbf{X})} } \) be the subset of all \( { \Phi\in{\textsf{C\!A}(\mathcal{A}^{\mathbb{M}})} } \) such that \( { \Phi({\mathbf{X}})\subseteq{\mathbf{X}} } \). Let \( { \mathfrak{M\scriptscriptstyle{\!e\!a\!s}}(\mathcal{A}^{\mathbb{M}}) } \) be the set of all probability measures on \( { \mathcal{A}^{\mathbb{M}} } \), and let \( { \mathfrak{M\scriptscriptstyle{\!e\!a\!s}}(\mathcal{A}^{\mathbb{M}};\Phi) } \) be the subset of Φ?invariant measures. If \( { {\mathbf{X}}\subset\mathcal{A}^{\mathbb{M}} } \), then let \( { \mathfrak{M\scriptscriptstyle{\!e\!a\!s}}({\mathbf{X}}) } \) be the set of probability measures μ with \( { {\textsf{supp}}(\mu)\subseteq{\mathbf{X}} } \), and define \( { \mathfrak{M\scriptscriptstyle{\!e\!a\!s}}({\mathbf{X}};\Phi) } \) in the obvious way.

Font conventions :

Upper case calligraphic letters (\( { {\mathcal{A}},{\mathcal{B}},{\mathcal{C}},\ldots } \)) denote finite alphabets or groups. Upper-case bold letters (\( { {\mathbf{A}},{\mathbf{B}},{\mathbf{C}},\ldots } \)) denote subsets of \( { \mathcal{A}^{\mathbb{M}} } \) (e.?g. subshifts), lowercase bold-faced letters (\( { {\mathbf{a}},{\mathbf{b}},{\mathbf{c}},\ldots } \)) denote elements of \( { \mathcal{A}^{\mathbb{M}} } \), and Roman letters (\( { a,b,c,\ldots } \)) are elements of \( { {\mathcal{A}} } \) or ordinary numbers. Lower-case sans-serif (\( { \ldots,{\textsf{m}},{\textsf{n}},{\textsf{p}} } \)) are elements of \( { {\mathbb{M}} } \), upper-case hollow font (\( { {\mathbb{U}},{\mathbb{V}},{\mathbb{W}},\ldots } \)) are subsets of \( { {\mathbb{M}} } \). Upper-case Greek letters (\( { \Phi,\Psi,\ldots } \)) are functions on \( { \mathcal{A}^{\mathbb{M}} } \) (e.?g. CA, block maps), and lower-case Greek letters (\( { \phi,\psi,\ldots } \)) are other functions (e.?g. local rules, measures.)

Acronyms in square brackets (e.?g. Topological Dynamics of Cellular Automata) indicate cross‐references to related entries in the Encyclopedia; these are listed at the end of this article.

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Acknowledgments

I would like to thank François Blanchard, Mike Boyle, Maurice Courbage, Doug Lind, Petr Kůrka, Servet Martínez, Kyewon Koh Park,Mathieu Sablik, Jeffrey Steif, and Marcelo Sobottka, who read draft versions of this article and made many invaluable suggestions, corrections, andcomments. (Any errors which remain are mine.) To Reem.

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Authors and Affiliations

  1. Department of Mathematics, Trent University, Peterborough, Canada

    Marcus Pivato

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  1. Marcus Pivato
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2012 Springer-Verlag

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Pivato, M. (2012). Ergodic Theory of Cellular Automata. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_62

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