Guy Barat: On flows arising from digital functions

This is a joint work with Pierre Liardet.

A $q-$multiplicative function is an unimodular sequence $(u(n))$ satisfying the conditions

$$u\left(\sum_{n=0}^\infty \varepsilon_nq^n\right)=\prod_{n=0}^\infty u(\varepsilon_nq^n)\;\;$$and$$\;\; u(0)=1.$$

A classical object related to the sequence $u$ is the dynamical system $(F(u),\sigma)$, where $\sigma$ is the shift operator and $F(u)$ the closed orbit of $u$ under its action. We study the structure of this dynamical system and especially give conditions under which it is minimal and/or ergodic. The case of $\alpha-$multiplicative functions (which is the natural analogon w.r.t. Ostrowski's systems of numeration) is also investigated. The main tools are skew products, essential values in the sense of Klaus Schmidt and appropriate compactifications of the set of the natural integers (odometers). Previous results of Liardet, Lesigne, and Mauduit are retrieved and extended.

Bibliography

1

Guy Barat and Pierre Liardet.
Dynamical systems originated in the ostrowski alpha-expansion.
(to appear in Publ. Math. Debrecen).

2

Emmanuel Lesigne, Christian Mauduit, and Brigitte Mossé.
Le théorème ergodique le long d'une suite $q$-multiplicative.
Compositio Math., 93(1):49-79, 1994.

3

Klaus Schmidt.
Cocycles on ergodic transformation groups.
Macmillan Company of India, Ltd., Delhi, 1977.
Macmillan Lectures in Mathematics, Vol. 1.

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