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SUMMARY:Manuel Stadlbauer (Universidade Federal do Rio de Janeiro)
DTSTART:20210521T081500Z
DTEND:20210521T094500Z
DTSTAMP:20260423T021827Z
UID:DSSUJ/29
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/DSSUJ/29/">A
  logarithmic law for continued fractions with sequentially restricted entr
 ies</a>\nby Manuel Stadlbauer (Universidade Federal do Rio de Janeiro) as 
 part of Dynamical systems seminar at the Jagiellonian University\n\n\nAbst
 ract\nNon-stationary shift spaces are basic models of sequential dynamical
  system who were intensively studied in order to construct symbolic models
  for ergodic automorphism (Vershik) or in the context of the isomorphism p
 roblem of shift spaces (Krieger). Recently\, the focus moved towards therm
 odynamic formalism and related questions. A fundamental tool of thermodyna
 mic formalism\, Ruelle's operator theorem\, has no immediate generalizatio
 n to the non-stationary setting as invariant functions intrinsically may n
 ot exist. However\, it is possible to establish geometric ergodicity for a
  family of ratios of operators. \n\nThis approach has applications to a cl
 assical problem in metric number theory. For a sequence $(\\alpha_n)$ conv
 erging to $\\infty$\, set \n\\[X_\\alpha := \\left\\{ x = \\frac{1}{x_1 + 
 \\frac{1}{x_2 + \\cdots} } :  x_n \\in \\mathbb N\, x_n \\geq  \\alpha_n \
 \hbox{ for all } n \\right\\}.\\] \nThat is\, $X$ is the subset of $[0\,1]
 $ such that  the $n$-th entry of the continued fraction expansion of each 
 element is bigger than or equal to $\\alpha_n$. In this setting\, for $\\a
 lpha_n \\gg n^{1+\\epsilon}$\, the  geometric ergodicity implies a law of 
 the iterated logarithm for square integrable functions from geometric ergo
 dicity. If\, in addition\, $(\\alpha_n)$ does not behave too wildly\, the 
 reference measure is absolutely contiunous with respect to the Hausdorff m
 easure (and the Hausdorff dimension is $1/2$). \\\\\n
LOCATION:https://researchseminars.org/talk/DSSUJ/29/
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