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SUMMARY:Victor Kleptsyn (Institut de Recherche Mathématique de Rennes)
DESCRIPTION:Rotation numbers of skew products and their dependence on a par
 ameter\nAbstract: Given a skew product over an ergodic transformation with
  the circle as the fiber\, one can define the associated (fiberwise) rotat
 ion number. If the skew product depends continuously on a parameter\, then
  so does the rotation number\; my talk with be devoted to the regularity o
 f this dependence.\nThis question is motivated\, in particular\, by the st
 udy of the discrete Schrödinger operator with dynamically defined potenti
 al. For such an operator\, the distribution function of the density of sta
 tes measure (DOS) is exactly the rotation number of the associated S^1-coc
 ycle as a function of energy as a parameter.\nI will present two results:\
 n1) The rotation number is (under very mild assumptions) log-Hölder\; thi
 s is a joint result with Anton Gorodetski (https://doi.org/10.1017/etds.20
 25.10195). This statement provides a dynamical viewpoint on the Craig-Simo
 n theorem\, stating the log-Hölder regularity for the DOS.\n2) It turns o
 ut that the increment of the rotation number can be expressed in terms of 
 invariant measures of the skew products (with the corresponding parameter 
 values). For the random dynamics with i.i.d. maps\, this gives a formula f
 or the increment of the rotation number in terms of (forward and backward)
  stationary measures of the dynamics. This integral formula explains the a
 nalogy between known results on the regularity of DOS and stationary measu
 res: the rotation number is at least as regular as the stationary measures
  (at least up to C^1 regularity). This is a joint work with Pedro Duarte a
 nd Anton Gorodetski (https://arxiv.org/abs/2512.00195).
URL:https://www.imperial.ac.uk/events/211274/victor-kleptsyn-institut-de-re
 cherche-mathematique-de-rennes/
DTSTART;TZID=Europe/London:20260630T110000
DTEND;TZID=Europe/London:20260630T120000
LOCATION:130\, Huxley Building\, South Kensington Campus\, Imperial College
  London\, London\, SW7 2AZ\, United Kingdom
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