150 research outputs found
Overlapping self-affine sets of Kakeya type
We compute the Minkowski dimension for a family of self-affine sets on the
plane. Our result holds for every (rather than generic) set in the class.
Moreover, we exhibit explicit open subsets of this class where we allow
overlapping, and do not impose any conditions on the norms of the linear maps.
The family under consideration was inspired by the theory of Kakeya sets.Comment: 27 pages, 1 figure. Submitted October 200
Measures with predetermined regularity and inhomogeneous self-similar sets
We show that if is a uniformly perfect complete metric space satisfying
the finite doubling property, then there exists a fully supported measure with
lower regularity dimension as close to the lower dimension of as we wish.
Furthermore, we show that, under the condensation open set condition, the lower
dimension of an inhomogeneous self-similar set coincides with the lower
dimension of the condensation set , while the Assouad dimension of is
the maximum of the Assouad dimensions of the corresponding self-similar set
and the condensation set . If the Assouad dimension of is strictly
smaller than the Assouad dimension of , then the upper regularity dimension
of any measure supported on is strictly larger than the Assouad dimension
of . Surprisingly, the corresponding statement for the lower regularity
dimension fails
Weak separation condition, Assouad dimension, and Furstenberg homogeneity
We consider dimensional properties of limit sets of Moran constructions
satisfying the finite clustering property. Just to name a few, such limit sets
include self-conformal sets satisfying the weak separation condition and
certain sub-self-affine sets. In addition to dimension results for the limit
set, we manage to express the Assouad dimension of any closed subset of a
self-conformal set by means of the Hausdorff dimension. As an interesting
consequence of this, we show that a Furstenberg homogeneous self-similar set in
the real line satisfies the weak separation condition. We also exhibit a
self-similar set which satisfies the open set condition but fails to be
Furstenberg homogeneous.Comment: 22 pages, 2 figure
Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension
A fundamental problem in the dimension theory of self-affine sets is the
construction of high-dimensional measures which yield sharp lower bounds for
the Hausdorff dimension of the set. A natural strategy for the construction of
such high-dimensional measures is to investigate measures of maximal Lyapunov
dimension; these measures can be alternatively interpreted as equilibrium
states of the singular value function introduced by Falconer. Whilst the
existence of these equilibrium states has been well-known for some years their
structure has remained elusive, particularly in dimensions higher than two. In
this article we give a complete description of the equilibrium states of the
singular value function in the three-dimensional case, showing in particular
that all such equilibrium states must be fully supported. In higher dimensions
we also give a new sufficient condition for the uniqueness of these equilibrium
states. As a corollary, giving a solution to a folklore open question in
dimension three, we prove that for a typical self-affine set in ,
removing one of the affine maps which defines the set results in a strict
reduction of the Hausdorff dimension
Existence of doubling measures via generalised nested cubes
Working on doubling metric spaces, we construct generalised dyadic cubes
adapting ultrametric structure. If the space is complete, then the existence of
such cubes and the mass distribution principle lead into a simple proof for the
existence of doubling measures. As an application, we show that for each
there is a doubling measure having full measure on a set of
packing dimension at most
- …