Slow continued fractions, Minkowski functions and the joint spectral radius

Giovanni Panti (Università degli Studi di Udine)

14-Jan-2021, 15:30-16:45 (3 years ago)

Abstract: Every unimodular partition of the real unit interval in m pieces gives rise to $2^m$ slow continued fraction maps. Many such maps have names (Farey fractions, ceiling fractions, even/odd fractions, ...), but most are nameless. Certain properties are commonly shared (for example, the validity of Lagrange's theorem), while other features are more delicate (the validity of the Serret theorem, the description of the unique a.c. invariant measure, the characterization of purely periodic points, ...).

Any slow continued fraction map determines a Minkowski function, namely the distribution function of the measure of maximal entropy. These Minkowski functions have a well-defined average Holder exponent (studied by many authors, and related to the dimension of the measure), as well as a least Holder exponent. The latter has the form log(m)/2*log(r), with r a quadratic irrational, the joint spectral radius of the iterated function system given by the inverse branches of the map.

It is plausible that every IFS with maps in $\mathrm{GL}(2,\mathbb Z)$ has algebraic joint spectral radius, but as far as we know this issue has not been settled. We show however, in joint work with Davide Sclosa, that this is indeed the case for IFSs over two maps in $\mathrm{SL}(2,\mathbb Z_{\geq 0})$.

dynamical systems

Audience: researchers in the topic


Bremen Online Dynamics Seminar

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