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Expansion subshifts of iterative systems\\
Petr K\r{u}rka
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An iterative system $(F,W)$ consists of a compact metric space $X$, a finite alphabet $A$, homeomorphisms $F=\{F_a: X \to X\}_{a\in A}$, and a regular open almost-cover $W=\{W_a:\; a\in A\}$ of $X$. This means $\overline{W_a}^{\circ} = W_a$ and $\bigcup_a \overline{W_a} = X$.
For $u\in A^*$ set $F_u = F_{u_{n-1}}\circ \cdots F_{u_0}$. A sequence $x_i\in X$ is a trajectory with itinerary $u\in A^N$, if $x_i\in W_{u_i}$ and $x_{i+1} = F_{u_i}(x_i)$. Denote by
$W_u = W_{u_0} \cap F^{-1}_{u_0}W_{u_1} \cap \cdots \cap F^{-1}_{u_{[0,n)}} W_{u_n}$
the set of points with itinerary $u$. The expansion subshift of $(F,W)$ is defined by
$$ S_{F,W} = \{u\in A^{N}:\; \forall n, W_{u_{[0,n)}}\neq \emptyset\}.$$
{\bf Proposition.} If $F_a: W_a \to X$ are expansions then there exists a continuous surjective function $\Phi:S_{F,W} \to X$ such that $$\{\Phi(u)\} = \bigcap_{n>0} \overline{W_{u_{[0,n)}}},\; u \in S_{F,W}.$$
{\bf Theorem.} $S_{F,W}$ is an SFT of order $m+1$ iff $\forall a\in A$, $\forall u\in A^m$,
$$W_u \cap F_aW_a \neq \emptyset \Rightarrow W_u \subseteq F_aW_a$$
{\bf Theorem.} $S_{F,W}$ is sofic iff there exists a partition $V=\{V_p:\; p\in B\}$ with\\
1. $V_p \cap W_a \neq \emptyset \Rightarrow V_p \subseteq W_a$\\
2. $V_p\subseteq W_a, V_q \cap F_aV_p \neq \emptyset \Rightarrow V_q \subseteq F_aV_p$.\\
In this case $S_{F,W}$ is the subshift of the labelled graph
$$ p \stackrel{a}{\to} q \Leftrightarrow V_p \subseteq W_a \;\&\;
V_q \subseteq F_aV_p $$ |
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