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Markov partitions and homoclinic points of algebraic \(\mathbb{Z}^d\)-actions. (English. Russian original) Zbl 0954.37008

Proc. Steklov Inst. Math. 216, 259-279 (1997); translation from Tr. Mat. Inst. Steklova 216, 265-284 (1997).
In order to present the main results of the paper we fix some background. One fixes a finite set \(A\), called alphabet and one defines the shift-action \(\rho\) of \(\mathbb{Z}^d\) on \(A^{\mathbb{Z}^{d}}\) by: \[ (\rho({\mathbf n},z))_{\mathbf m}=z_{{\mathbf m}+{\mathbf n}}, \forall {\mathbf n}\in\mathbb{Z}^{d}, z=(z_{\mathbf m})\in A^{\mathbb{Z}^{d}} \] In this context one defines a closed, shift-invariant set \(Z\subset A^{\mathbb{Z}^{d}}\), called shift of finite type. A closed, shift-invariant set \(Z\subset A^{\mathbb{Z}^{d}}\) is sofic if there exists a finite set \(A'\) , a shift of finite type \(Z'\subset {A'}^{\mathbb{Z}^{d}}\), and a continuous surjective map \(\phi:Z'\to Z\) such that \(\phi\cdot \rho'({\mathbf n},z)=\rho({\mathbf n},\phi(z))\), for every \(z\in Z'\) and \({\mathbf n}\in \mathbb{Z}^{d}\). If \(T:{\mathbf n}\mapsto T^{\mathbf n}\) is a continuous \(\mathbb{Z}^d\)-action on a compact space \(X\), \(A\) is an alphabet, \(\rho\) the shift-action of \(\mathbb{Z}^d\) on \(A^{\mathbb{Z}^{d}}\) and \(Y\subset A^{\mathbb{Z}^{d}}\) is a closed, shift-invariant subset, then \((Y,\rho_{|Y})\) is called a symbolic cover of \((X,T)\) if there exists a continuous, surjective map \(\phi:Y\to X\) with: \[ \phi\cdot \rho_{|Y}({\mathbf n}, \cdot)=T^{\mathbf n}\phi, \forall, {\mathbf n}\in\mathbb{Z}^{d} \] A symbolic cover \(Y\) of \((X,T)\) is called of equal entropy if the topological entropies \(h(\rho_{|Y})\) and \(h(T)\) coincide.
The authors restrict themselves to \(\mathbb{Z}^{d}\)-actions by automorphisms of compact Abelian groups, and call such an action algebraic \(\mathbb{Z}^{d}\)-action. They prove that expansive algebraic \(\mathbb{Z}^{d}\)-actions with completely positive entropy have symbolic covers of equal topological entropy. These symbolic covers are constructed by means of homoclinic points of these actions. For the particular case \(d=1\) it is shown that these symbolic covers are sofic shifts. For \(d\geq 2\) only for particular examples the analogous property is proved.
For the entire collection see [Zbl 0884.00028].

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B10 Symbolic dynamics
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