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Factors of Toeplitz flows and other almost 1-1 extensions over group rotations. (English) Zbl 1023.37004

Symbolic extensions of minimal flows are studied. The main theorems proved are the followings:

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,R ˜) of (G,R) such that (Y,S) is an almost 1-1 extension of (H,R ˜).

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).

To prove this theorem, the authors use return maps.

Using the above theorems, they characterize factors of Toeplitz flow:

Theorem. A dynamical system (X,T) is a factor of Toeplitz flow if and only if

1. (X,T) is minimal,

2. (X,T) is almost 1-1 extension of odometer,

3. (X,T) has symbolic extension.

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 .

37B05Transformations and group actions with special properties
37B10Symbolic dynamics