Random Fibonacci sequences
Introduction to Ergodic Theory and Additive Combinatorics
August 26, 2008 04:10 PM to 05:00 PM
Speakers:
de la Rue, Thierry
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Abstract: |
[talk based on a series of joint works with Élise janvresse (Rouen) and Benoît Rittaud (Paris 13 and Tours)]
Random Fibonacci sequences are given by their first two terms F_0 and F_1,
and one of the following stochastic inductions:
F_(n+1) = F_n +/- F_(n-1) (linear random Fibonacci)
or
F_(n+1) = | F_n +/- F_(n-1) | (non-linear random Fibonacci),
where the +/- signs are given by an i.i.d. Bernoulli process of parameter
p in [0,1] (p is the probability of a +). We are mainly interested in the
exponential growth (largest Lyapunov exponent) of these sequences. We show
that this exponents can in both cases be computed as an integral with
respect to some explicit probability measure inductively defined on
Stern-Brocot intervals. This allows us to study the variations of the
exponent with respect to the parameter p. In particular, we can show that
this exponent is positive for any p>0 in the linear case, whereas in the
non-linear case the exponential growth holds only when p>1/3.
To get the integral formula, we develop an original method whose main
ingredients are
- a reduction process of the random Fibonacci sequence, allowed by some
nontrivial identities satisfied by the 2X2 matrices which are involved in
the
induction;
- elementary properties of continued fraction expansion and Stern-Brocot
intervals.
This method can be extended to random Fibonacci sequences with
multiplicative coefficient:
F_(n+1) = lambda F_n +/- F_(n-1) (linear case)
or
F_(n+1) = | lambda F_n +/- F_(n-1) | (non-linear case),
where lambda is of the form lambda = lambda_k = 2 cos pi/k
(k=2,3,4,...)
In these cases, we have to introduce generalizations of Stern-Brocot
intervals and work with so-called Rosen continued fraction expansions. |
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