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Some constructions of strictly ergodic non-regular Toeplitz flows. (English) Zbl 0810.28009
The authors are interested in the strict ergodicity, regularity and cyclic approximation of Toeplitz flows and they obtain results and provide examples relevant to these issues.
Let \(A\) be a finite set and form the shift dynamical system \(\Omega= A^ \mathbb{Z}\) with shift \(\sigma\). A sequence \(\nu\in \Omega\) is called a Toeplitz sequence if it satisfies \[ \forall n\in \mathbb{Z} \exists p\geq 2\;\forall k\in \mathbb{Z}\;\nu(n+ kp)= \nu(n) \] and \(\nu\) is not a periodic sequence. A Toeplitz flow is a dynamical system \((\overline{O(\nu)},\sigma)\), where \(\nu\) is a Toeplitz sequence. Every Toeplitz flow is minimal and it is strictly ergodic if it admits exactly one invariant probability measure.
If we add an additional symbol \(\{\infty\}\) to \(A\) to get \(\overline A\) and form \(\overline\Omega= \overline A^ \mathbb{Z}\) then any Toeplitz sequence in \(\Omega\) can be expressed as \(\lim_{n\to\infty} B^ \infty_ n\) in \(\overline\Omega\), where \(B_ n\) is a word in \(\overline A^ \mathbb{Z}\), length of \(B_ n= p_ n\), \(p_ n\) is increasing, and (1) \(p_ n\geq 2\) with \(p_ n| p_{n+1}\), (2) \(B_{n+1}(j)= B_ n(i)\) whenever \(j\equiv i\text{ mod}(p_ n)\) and \(B_ n(i)\neq \infty\), (3) \(p_ n\) is the least period of \(B^ \infty_ n\). The authors use this construction to obtain a necessary and sufficient condition for a Toeplitz flow to be strictly ergodic.
In the second section it is shown by example that the regularity: \[ d_ n(\nu)= {1\over p_ n} (\#\{i: B_ n(i)\neq \infty, 0\leq i< p_ n\}), \] is not a topological invariant of the flow. They develop a notion of eventual regularity which is an “eventual invariant”.
In the third section the authors construct Toeplitz flows as group extensions over the maximal equicontinuous factor. This class admits good cyclic approximations which allows the authors to obtain spectral information.

MSC:
28D10 One-parameter continuous families of measure-preserving transformations
54H20 Topological dynamics (MSC2010)
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