Event Type:

Number Theory Seminar

Date/Time:

Tuesday, October 31, 2017 - 16:00 to 17:00

Location:

STAG 112

Local Speaker:

Abstract:

Abstract: In a letter to Laplace in 1812, Gauss gave the explicit formula for the invariant measure associated to regular continued fractions. To this day, we do not know how Gauss found the measure. With P. Arnoux, we give a heuristic method for determining such measures. To any ``expanding" piecewise differentiable function of the unit interval to itself, f:I->I, we associate an IFS (iterated function system) which has a fixed point K in the space of compact subsets of I x R. Lebesgue measure, m, is invariant for a naturally induced map F on K. When m(K)>0 and m( K\ F(K) ) = 0, integration along fibers gives an invariant measure for f. This is probably not how Gauss found his measure, but it does succeed for the continued fraction map and many more.