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### Open Access Journals

JMD

We show that there exist minimal interval-exchange transformations
with an ergodic measure whose Hausdorff dimension is arbitrarily
small, even 0. We will also show that in particular cases one can
bound the Hausdorff dimension between $\frac{1}{2r+4}$ and
$\frac{1}{r}$ for any r greater than 1.

JMD

We prove that the Birkhoff pointwise ergodic theorem and the Oseledets
multiplicative ergodic theorem hold for every flat surface in almost every
direction. The proofs rely on the strong law of large numbers, and on
recent rigidity results for the action of the upper triangular subgroup of
$SL(2,\mathbb R)$ on the moduli space of flat surfaces. Most of the results also use
a theorem about continuity of splittings of the Kontsevich-Zorich cocycle
recently proved by S. Filip.

JMD

We prove that there exists an interval exchange transformation
and a point so that the orbit of the point equidistributes according to a
non-ergodic measure. That is, it is possible for a non-ergodic measure to
arise from the Krylov-Bogolyubov construction of invariant measures for an
interval exchange transformation.

JMD

We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodic properties of translation flow than hyperbolic geometry.

JMD

We discuss discrete one-dimensional Schrödinger operators whose
potentials are generated by an invertible ergodic transformation
of a compact metric space and a continuous real-valued sampling
function. We pay particular attention to the case where the
transformation is a minimal interval-exchange transformation.
Results about the spectral type of these operators are
established. In particular, we provide the first examples of
transformations for which the associated Schrödinger operators
have purely singular spectrum for every nonconstant continuous
sampling function.

JMD

We prove that the set of bounded geodesics in Teichmüller space
is a winning set for Schmidt's game. This is a notion of largeness in
a metric space that can apply to measure $0$ and meager sets. We prove
analogous closely related results on any Riemann surface, in any
stratum of quadratic differentials, on any Teichmüller disk and
for intervals exchanges with any fixed irreducible permutation.

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