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SUMMARY:Margaret Stawiska-Friedland\\ (American Mathematical Society/Mathe
 matical Reviews\, USA)
DTSTART:20210511T130000Z
DTEND:20210511T140000Z
DTSTAMP:20260422T201527Z
UID:CAvid/40
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAvid/40/">A
  potential-theoretic characterization of polynomials in holomorphic dynami
 cs in one  variable}</a>\nby Margaret Stawiska-Friedland\\ (American Mathe
 matical Society/Mathematical Reviews\, USA) as part of CAvid: Complex Anal
 ysis video seminar\n\nLecture held in N/A.\n\nAbstract\nIn 1960s Hans Brol
 in initiated systematic application of potential-theoretic methods in the 
 dynamics of holomorphic maps. Among other things\, he proved the now-famou
 s equidistribution theorem: for a complex polynomial $f$ of degree greater
  than $1$ the preimages\, under successive iterates of $f$\, of a Dirac me
 asure at an arbitrary point of the complex plane (except at most two so-ca
 lled exceptional points) converge weakly to the equilibrium measure (with 
 pole at infinity) for the Julia set $J_f$ of $f$. To a general rational ma
 p $f$ of degree $d \\geq 2$ on the Riemann sphere $\\mathbb{CP}^1$ one can
  associate another measure $\\mu$\, called the balanced measure. It is sup
 ported on the Julia set for $f$ and satisfies  $f*\\mu=d \\cdot \\mu$. Sin
 ce it also can be obtained as the limit of  preimages of quite general pro
 babilistic measures on $\\mathbb{CP}^1$ (thanks to the results of M. Lyubi
 ch and independently Freire-Lopes-R. Ma\\~ne from 1980s)\, a question aris
 es whether it always equals the equilibrium measure for $J_f$ (when the la
 tter notion makes sense). Several mathematicians noticed  that equality of
  these two measures (under suitable assumptions on $f$) implies that $f$ i
 s a polynomial. However\, all  ``proofs''  of this implication from  befor
 e 1990s contained gaps. The proof  by S. Lalley from 1992 was fully succes
 sful\, but it was based on the theory of Brownian motions. In this talk\, 
 I will present a  general version  of this implication  with a proof using
  mainly classical and weighted potential theory: Let $f:  \\mathbb{CP}^1 \
 \to \\mathbb{CP}^1$ be a rational function of degree $d \\geq 2$ whose Jul
 ia set does not contain the point $\\infty$. The following are equivalent:
  (i) $f \\circ f$ is a polynomial\; (ii) the balanced measure for $f$ and 
 the  equilibrium measure for the Julia set $J_f$  with pole at infinity ar
 e equal. This  is joint work with Y\\^usuke Okuyama from Kyoto Institute o
 f Technology.\n
LOCATION:https://researchseminars.org/talk/CAvid/40/
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