Textile systems for endomorphisms and automorphisms of the shift.

*(English)*Zbl 0845.54031
Mem. Am. Math. Soc. 546, 215 p. (1995).

This memoir deals with textile systems. This notion is used to identify \((X, \varphi)\), where \(\varphi\) is an expansive automorphism of a subshift \((X, \sigma)\), with some topological class of subshifts.

A textile \(T\) is an ordered pair of graph-homomorphisms \(T= (p,q: \Gamma\to G)\) \((\Gamma\), \(G\) graphs, with some constraints). Let \((X_\Gamma, \sigma_\Gamma)\) and \((X_G, \sigma_G)\) be topological Markov shifts defined by the graphs \(\Gamma\) and \(G\), and \(\xi, \eta: (X_\Gamma, \sigma_\Gamma)\to (X_G, \sigma_G)\) be factor maps given by \(p\) and \(q\). Define two shifts \((X_T, \sigma_T)\) and \((Z_T, \rho_T)\) by: Let \(X_0= X_G\) and \(Z_0= X_\Gamma\), set \(X_n= \xi (Z_{n-1}) \cap \eta (Z_{n-1})\) and \(Z_n= \xi^{-1} (X_{n-1}) \cap \eta^{-1} (X_{n-1})\), and let \(X_T= \bigcap X_n\), \(\sigma_T= \sigma_G|X_T\) and \(Z_T= \bigcap Z_n\), \(\rho= \sigma_\Gamma |Z_T\). Then we have two factor maps \(\xi_T\), \(\eta_T\) deduced from \(\xi\), \(\eta\), and we can set \(\varphi= \eta_T \xi_T^{-1}\) (similarly for one-sided shifts).

Given a textile \(T= (p, g: \Gamma, G)\), one can define a dual system. The graphs \(G^T\), \(\Gamma^T\) are defined by: the arc-set of \(G^T\) is the vertex-set of \(\Gamma\) and its vertex-set is the vertex-set of \(G\) and the initial and terminal vertices of an arc \(a\in \Gamma\) are given by \(p(a)\) and \(q(a)\). The graph \(\Gamma^T\) is given similarly using the arc-sets of \(\Gamma\) and \(G\). Two new graph homomorphisms are done by initial and terminal vertice maps \(p^T= i_\Gamma\), \(q^T= i_G\). The system \(T^*= (p^T, q^T: \Gamma^T\to G^T)\) is said to be a dual to \(T\).

These notions permit to classify the classes of topological conjugates in case of expansiveness. Here is an example. The textile system \(T\) is ‘1-1’ iff both \(\xi_T\), \(\eta_T\) are ‘1-1’, then \(\varphi\) is an automorphism of \(x_T\), \(\sigma_T\). If \(T\) is ‘1-1’ then \(\varphi_T\) is expansive iff \(T^*\) is ‘1-1’. If both dual systems are ‘1-1’ then \((X_T, \varphi_T)\) is topologically conjugate to \((X_{T^*}, \sigma_{T^*})\). Then the study of automorphisms \(\varphi^k \sigma^n\) follows.

Next, the author deals with classes of automorphisms of one-sided topological Markov shifts. Every such automorphism is equal to \(\varphi_T\) for some one sided ‘1-1’ textile system with both \(p\), \(q\) left resolving. Then sofic and LR textile systems are studied and notions of ‘similarity’ and ‘weak similarity’ are introduced and studied.

A textile \(T\) is an ordered pair of graph-homomorphisms \(T= (p,q: \Gamma\to G)\) \((\Gamma\), \(G\) graphs, with some constraints). Let \((X_\Gamma, \sigma_\Gamma)\) and \((X_G, \sigma_G)\) be topological Markov shifts defined by the graphs \(\Gamma\) and \(G\), and \(\xi, \eta: (X_\Gamma, \sigma_\Gamma)\to (X_G, \sigma_G)\) be factor maps given by \(p\) and \(q\). Define two shifts \((X_T, \sigma_T)\) and \((Z_T, \rho_T)\) by: Let \(X_0= X_G\) and \(Z_0= X_\Gamma\), set \(X_n= \xi (Z_{n-1}) \cap \eta (Z_{n-1})\) and \(Z_n= \xi^{-1} (X_{n-1}) \cap \eta^{-1} (X_{n-1})\), and let \(X_T= \bigcap X_n\), \(\sigma_T= \sigma_G|X_T\) and \(Z_T= \bigcap Z_n\), \(\rho= \sigma_\Gamma |Z_T\). Then we have two factor maps \(\xi_T\), \(\eta_T\) deduced from \(\xi\), \(\eta\), and we can set \(\varphi= \eta_T \xi_T^{-1}\) (similarly for one-sided shifts).

Given a textile \(T= (p, g: \Gamma, G)\), one can define a dual system. The graphs \(G^T\), \(\Gamma^T\) are defined by: the arc-set of \(G^T\) is the vertex-set of \(\Gamma\) and its vertex-set is the vertex-set of \(G\) and the initial and terminal vertices of an arc \(a\in \Gamma\) are given by \(p(a)\) and \(q(a)\). The graph \(\Gamma^T\) is given similarly using the arc-sets of \(\Gamma\) and \(G\). Two new graph homomorphisms are done by initial and terminal vertice maps \(p^T= i_\Gamma\), \(q^T= i_G\). The system \(T^*= (p^T, q^T: \Gamma^T\to G^T)\) is said to be a dual to \(T\).

These notions permit to classify the classes of topological conjugates in case of expansiveness. Here is an example. The textile system \(T\) is ‘1-1’ iff both \(\xi_T\), \(\eta_T\) are ‘1-1’, then \(\varphi\) is an automorphism of \(x_T\), \(\sigma_T\). If \(T\) is ‘1-1’ then \(\varphi_T\) is expansive iff \(T^*\) is ‘1-1’. If both dual systems are ‘1-1’ then \((X_T, \varphi_T)\) is topologically conjugate to \((X_{T^*}, \sigma_{T^*})\). Then the study of automorphisms \(\varphi^k \sigma^n\) follows.

Next, the author deals with classes of automorphisms of one-sided topological Markov shifts. Every such automorphism is equal to \(\varphi_T\) for some one sided ‘1-1’ textile system with both \(p\), \(q\) left resolving. Then sofic and LR textile systems are studied and notions of ‘similarity’ and ‘weak similarity’ are introduced and studied.

Reviewer: T.Nowicki (Warszawa)

##### MSC:

54H20 | Topological dynamics (MSC2010) |

37B99 | Topological dynamics |