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SUMMARY:Peter Varju (University of Cambridge)
DTSTART:20210322T161500Z
DTEND:20210322T173000Z
DTSTAMP:20260423T053138Z
UID:NEDNT/18
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NEDNT/18/">O
 n the dimension of self-similar measures</a>\nby Peter Varju (University o
 f Cambridge) as part of New England Dynamics and Number Theory Seminar\n\n
 Lecture held in Online.\n\nAbstract\nLet $f_1$\,…\,$f_n$ be a collection
  of contracting similarities on $\\mathbb{R}$\, and let $p_1$\,…\,$p_n$ 
 be a probability vector. There is a unique probability measure mu on $\\ma
 thbb{R}$ that satisfies the identity\n$\\mu = p_1 f_1(\\mu) + … + p_n f_
 n(\\mu)$.\nThis measure is called self-similar. The maps $f_1$\,…\,$f_n$
  are said to satisfy the no exact overlaps condition if they generate a fr
 ee semigroup (i.e. all compositions are distinct). Under this condition\, 
 the dimension of mu is conjectured to be the minimum of 1 and the ratio of
  the entropy of $p_1$\,…\,$p_n$ and the average logarithmic contraction 
 factor of the $f_i$. This conjecture has been recently established in some
  special cases\, including when $n=2$ and $f_1$ and $f_2$ have the same co
 ntraction factor. In the talk I will discuss recent progress by Ariel Rapa
 port and myself in the case $n=3$. In this case new difficulties arise as 
 was demonstrated by recent examples of Baker and Barany\, Kaenmaki of IFS
 ’s with arbitrarily weak separation properties.\n
LOCATION:https://researchseminars.org/talk/NEDNT/18/
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