29 research outputs found
Duality Theorems in Ergodic Transport
We analyze several problems of Optimal Transport Theory in the setting of
Ergodic Theory. In a certain class of problems we consider questions in Ergodic
Transport which are generalizations of the ones in Ergodic Optimization.
Another class of problems is the following: suppose is the shift
acting on Bernoulli space , and, consider a fixed
continuous cost function . Denote by the set
of all Borel probabilities on , such that, both its and
marginal are -invariant probabilities. We are interested in the
optimal plan which minimizes among the probabilities on
.
We show, among other things, the analogous Kantorovich Duality Theorem. We
also analyze uniqueness of the optimal plan under generic assumptions on .
We investigate the existence of a dual pair of Lipschitz functions which
realizes the present dual Kantorovich problem under the assumption that the
cost is Lipschitz continuous. For continuous costs the corresponding
results in the Classical Transport Theory and in Ergodic Transport Theory can
be, eventually, different.
We also consider the problem of approximating the optimal plan by
convex combinations of plans such that the support projects in periodic orbits
On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions
We consider the "Mandelbrot set" for pairs of complex linear maps,
introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and
others. It is defined as the set of parameters in the unit disk such
that the attractor of the IFS is
connected. We show that a non-trivial portion of near the imaginary axis is
contained in the closure of its interior (it is conjectured that all non-real
points of are in the closure of the set of interior points of ). Next we
turn to the attractors themselves and to natural measures
supported on them. These measures are the complex analogs of
much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os
and Garsia, we demonstrate how certain classes of complex algebraic integers
give rise to singular and absolutely continuous measures . Next we
investigate the Hausdorff dimension and measure of , for
in the set , for Lebesgue-a.e. . We also obtain partial results on
the absolute continuity of for a.e. of modulus greater
than .Comment: 22 pages, 5 figure
Wavelets techniques for pointwise anti-Holderian irregularity
In this paper, we introduce a notion of weak pointwise Holder regularity,
starting from the de nition of the pointwise anti-Holder irregularity. Using
this concept, a weak spectrum of singularities can be de ned as for the usual
pointwise Holder regularity. We build a class of wavelet series satisfying the
multifractal formalism and thus show the optimality of the upper bound. We also
show that the weak spectrum of singularities is disconnected from the casual
one (denoted here strong spectrum of singularities) by exhibiting a
multifractal function made of Davenport series whose weak spectrum di ers from
the strong one
Flatness is a Criterion for Selection of Maximizing Measures
For a full shift with Np+1 symbols and for a non-positive potential, locally
proportional to the distance to one of N disjoint full shifts with p symbols,
we prove that the equilibrium state converges as the temperature goes to 0. The
main result is that the limit is a convex combination of the two ergodic
measures with maximal entropy among maximizing measures and whose supports are
the two shifts where the potential is the flattest. In particular, this is a
hint to solve the open problem of selection, and this indicates that flatness
is probably a/the criterion for selection as it was conjectured by A.O. Lopes.
As a by product we get convergence of the eigenfunction at the log-scale to a
unique calibrated subaction