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SUMMARY:Tom Kempton (University of Manchester)
DTSTART:20210119T133000Z
DTEND:20210119T143000Z
DTSTAMP:20260423T052922Z
UID:OWNS/29
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OWNS/29/">Be
 rnoulli Convolutions and Measures on the Spectra of Algebraic Integers</a>
 \nby Tom Kempton (University of Manchester) as part of One World Numeratio
 n seminar\n\n\nAbstract\nGiven an algebraic integer $\\beta$ and alphabet 
 $A=\\{-1\,0\,1\\}$\, the spectrum of $\\beta$ is the set \n$$\\Sigma(\\bet
 a) :=\\bigg\\{\\sum_{i=1}^n a_i\\beta^i : n\\in\\mathbb N\, a_i\\in A\\big
 g\\}.$$\nIn the case that $\\beta$ is Pisot one can study the spectrum of 
 $\\beta$ dynamically using substitutions or cut and project schemes\, and 
 this allows one to see lots of local structure in the spectrum. There are 
 higher dimensional analogues for other algebraic integers.\n\nIn this talk
  we will define a random walk on the spectrum of $\\beta$ and show how\, w
 ith appropriate renormalisation\, this leads to an infinite stationary mea
 sure on the spectrum. This measure has local structure analagous to that o
 f the spectrum itself. Furthermore\, this measure has deep links with the 
 Bernoulli convolution\, and in particular new criteria for the absolute co
 ntinuity of Bernoulli convolutions can be stated in terms of the ergodic p
 roperties of these measures.\n
LOCATION:https://researchseminars.org/talk/OWNS/29/
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