50 research outputs found
Mixing of asymmetric logarithmic suspension flows over interval exchange transformations
We consider suspension flows built over interval exchange transformations
with the help of roof functions having an asymmetric logarithmic singularity.
We prove that such flows are strongly mixing for a full measure set of interval
exchange transformations
Ergodic properties of infinite extensions of area-preserving flows
We consider volume-preserving flows on , where is a closed connected surface of genus and
has the form , where is a locally Hamiltonian flow
of hyperbolic periodic type on and is a smooth real valued function on
. We investigate ergodic properties of these infinite measure-preserving
flows and prove that if belongs to a space of finite codimension in
, then the following dynamical dichotomy holds: if
there is a fixed point of on which does not
vanish, then is ergodic, otherwise, if
vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial
extension . The proof of this result exploits the
reduction of to a skew product automorphism over
an interval exchange transformation of periodic type. If there is a fixed point
of on which does not vanish, the reduction
yields cocycles with symmetric logarithmic singularities, for which we prove
ergodicity.Comment: 57 pages, 4 picture
A temporal Central Limit Theorem for real-valued cocycles over rotations
We consider deterministic random walks on the real line driven by irrational
rotations, or equivalently, skew product extensions of a rotation by
where the skewing cocycle is a piecewise constant mean zero function with a
jump by one at a point . When is badly approximable and
is badly approximable with respect to , we prove a Temporal Central
Limit theorem (in the terminology recently introduced by D.Dolgopyat and
O.Sarig), namely we show that for any fixed initial point, the occupancy random
variables, suitably rescaled, converge to a Gaussian random variable. This
result generalizes and extends a theorem by J. Beck for the special case when
is quadratic irrational, is rational and the initial point is
the origin, recently reproved and then generalized to cover any initial point
using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig
(Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use
renormalization, but in order to treat irrational values of , instead of
geometric arguments, we use the renormalization associated to the continued
fraction algorithm and dynamical Ostrowski expansions. This yields a suitable
symbolic coding framework which allows us to reduce the main result to a CLT
for non homogeneous Markov chains.Comment: a few typos corrected, 28 pages, 4 figure