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On the evolution of continued fraction in a fixed quadratic field
Thu 28 February 2013, 15:00
Ergodic Theory and Dynamical Systems
Organisers: Andrew Ferguson, Thomas Jordan
ABSTRACT
It is well known that quadratic irrationals are characterized as
the numbers whose continued fraction expansion is eventually periodic.
Each quadratic irrational exhibits then a certain statistics in its period
(for example, one can measure the frequency of the digit 1 in the period).
I will present results regarding the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the `normal' statistics given by the Gauss-Kuzmin measure.
Our proof exploits a classical connection between the continued fraction
expansion and dynamics on the modular surface. I will review this connection
and explain the proof of an equivalent equidistribution result on a
S-arithmetic homogeneous space. Our proof is a simple example of dynamical analysis on S-arithmetic homogeneous spaces with application to number theory.
These results are from a joint work with Uri Shapira.