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SUMMARY:Giovanni Panti (Università degli Studi di Udine)
DTSTART:20210114T153000Z
DTEND:20210114T164500Z
DTSTAMP:20260423T021435Z
UID:BODS/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/BODS/1/">Slo
 w continued fractions\, Minkowski functions and the joint spectral radius<
 /a>\nby Giovanni Panti (Università degli Studi di Udine) as part of Breme
 n Online Dynamics Seminar\n\n\nAbstract\nEvery unimodular partition of the
  real unit interval in m pieces gives\nrise to $2^m$ slow continued fracti
 on maps. Many such maps have names\n(Farey fractions\, ceiling fractions\,
  even/odd fractions\, ...)\, but most\nare nameless. Certain properties ar
 e commonly shared (for example\, the\nvalidity of Lagrange's theorem)\, wh
 ile other features are more delicate\n(the validity of the Serret theorem\
 , the description of the unique a.c.\ninvariant measure\, the characteriza
 tion of purely periodic points\, ...).\n\nAny slow continued fraction map 
 determines a Minkowski function\, namely\nthe distribution function of the
  measure of maximal entropy. These\nMinkowski functions have a well-define
 d average Holder exponent (studied\nby many authors\, and related to the d
 imension of the measure)\, as well\nas a least Holder exponent. The latter
  has the form log(m)/2*log(r)\,\nwith r a quadratic irrational\, the joint
  spectral radius of the iterated\nfunction system given by the inverse bra
 nches of the map.\n\nIt is plausible that every IFS with maps in $\\mathrm
 {GL}(2\,\\mathbb Z)$ has algebraic joint\nspectral radius\, but as far as 
 we know this issue has not been settled.\nWe show however\, in joint work 
 with Davide Sclosa\, that this is indeed\nthe case for IFSs over two maps 
 in $\\mathrm{SL}(2\,\\mathbb Z_{\\geq 0})$.\n
LOCATION:https://researchseminars.org/talk/BODS/1/
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