1,738 research outputs found
First and second moments for self-similar couplings and Wasserstein distances
We study aspects of the Wasserstein distance in the context of self-similar
measures. Computing this distance between two measures involves minimising
certain moment integrals over the space of \emph{couplings}, which are measures
on the product space with the original measures as prescribed marginals. We
focus our attention on self-similar measures associated to equicontractive
iterated function systems satisfying the open set condition and consisting of
two maps on the unit interval. We are particularly interested in understanding
the restricted family of \emph{self-similar} couplings and our main achievement
is the explicit computation of the 1st and 2nd moment integrals for such
couplings. We show that this family is enough to yield an explicit formula for
the 1st Wasserstein distance and provide non-trivial upper and lower bounds for
the 2nd Wasserstein distance.Comment: 14 pages, 3 figure
Remarks on the analyticity of subadditive pressure for products of triangular matrices
We study Falconer's subadditive pressure function with emphasis on
analyticity. We begin by deriving a simple closed form expression for the
pressure in the case of diagonal matrices and, by identifying phase transitions
with zeros of Dirichlet polynomials, use this to deduce that the pressure is
piecewise real analytic. We then specialise to the iterated function system
setting and use a result of Falconer and Miao to extend our results to include
the pressure for systems generated by matrices which are simultaneously
triangularisable. Our closed form expression for the pressure simplifies a
similar expression given by Falconer and Miao by reducing the number of
equations needing to be solved by an exponential factor. Finally we present
some examples where the pressure has a phase transition at a non-integer value
and pose some open questions.Comment: 10 pages, 1 figure, to appear in Monatshefte f\"ur Mathemati
Assouad type dimensions and homogeneity of fractals
We investigate several aspects of the Assouad dimension and the lower
dimension, which together form a natural `dimension pair'. In particular, we
compute these dimensions for certain classes of self-affine sets and
quasi-self-similar sets and study their relationships with other notions of
dimension, like the Hausdorff dimension for example. We also investigate some
basic properties of these dimensions including their behaviour regarding unions
and products and their set theoretic complexity.Comment: 40 pages, 6 figure
Inhomogeneous self-similar sets and box dimensions
We investigate the box dimensions of inhomogeneous self-similar sets.
Firstly, we extend some results of Olsen and Snigireva by computing the upper
box dimensions assuming some mild separation conditions. Secondly, we
investigate the more difficult problem of computing the lower box dimension. We
give some non-trivial bounds and provide examples to show that lower box
dimension behaves much more strangely than the upper box dimension, Hausdorff
dimension and packing dimension.Comment: To appear in Studia Mathematica, 20 pages, 4 figure
On the packing dimension of box-like self-affine sets in the plane
We consider a class of planar self-affine sets which we call "box-like". A
box-like self-affine set is the attractor of an iterated function system (IFS)
of affine maps where the image of the unit square, [0,1]^2, under arbitrary
compositions of the maps is a rectangle with sides parallel to the axes. This
class contains the Bedford-McMullen carpets and the generalisations thereof
considered by Lalley-Gatzouras, Bara\'nski and Feng-Wang as well as many other
sets. In particular, we allow the mappings in the IFS to have non-trivial
rotational and reflectional components. Assuming a rectangular open set
condition, we compute the packing and box-counting dimensions by means of a
pressure type formula based on the singular values of the maps.Comment: 15 pages, 4 figure
On the dimensions of a family of overlapping self-affine carpets
We consider the dimensions of a family of self-affine sets related to the
Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and
then randomise the translation vectors with the stipulation that the column
structure is preserved. As such, we maintain one of the key features in the
Bedford-McMullen set up in that alignment causes the dimensions to drop from
the affinity dimension. We compute the Hausdorff, packing and box dimensions
outside of a small set of exceptional translations, and also for some explicit
translations even in the presence of overlapping. Our results rely on, and can
be seen as a partial extension of, M. Hochman's recent work on the dimensions
of self-similar sets and measures.Comment: 17 pages, 5 figures, to appear in Ergodic Th. Dynam. Sys
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