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Zbl 1023.37004
Downarowicz, T.; Durand, F.
Factors of Toeplitz flows and other almost 1-1 extensions over group rotations. (English)
[J] Math. Scand. 90, No.1, 57-72 (2002). ISSN 0025-5521

Symbolic extensions of minimal flows are studied. The main theorems proved are the followings: \par Theorem. Let $(X,T)$ be a minimal almost 1-1 extension of odometer $(G,R)$, and $(Y,S)$ be a factor of $(X,T)$. Then there exists a factor $(H,\widetilde R)$ of $(G,R)$ such that $(Y,S)$ is an almost 1-1 extension of $(H,\widetilde R)$. \par Theorem. Let $(Z,T)$ be a minimal flow and $(X,S)$ be a symbolic extension of $(Z,T)$. Then there exists a symbolic almost 1-1 extension $(Y,S)$ of $(Z,T)$. \par To prove this theorem, the authors use return maps. \par Using the above theorems, they characterize factors of Toeplitz flow: \par Theorem. A dynamical system $(X,T)$ is a factor of Toeplitz flow if and only if \par 1. $(X,T)$ is minimal, \par 2. $(X,T)$ is almost 1-1 extension of odometer, \par 3. $(X,T)$ has symbolic extension. \par Theorem. A dynamical system $(X,T)$ is a minimal almost 1-1 extension of $(G,R)$ if and only if it is isomorphic to $(X_f,S)$, where $f$ is invariant under no rotation, where $X_f$ is the shift closure of $(f(n))_{n\in \bbfZ}$.
[Makoto Mori (Tokyo)]

MSC 2000:: *37B05 Transformations and group actions with special properties
37B10 Symbolic dynamics

Keywords: Toeplitz flow; topological factor; almost 1-1 extension; symbolic extension; minimal flows

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