5,916 research outputs found
Self-affine Manifolds
This paper studies closed 3-manifolds which are the attractors of a system of
finitely many affine contractions that tile . Such attractors are
called self-affine tiles. Effective characterization and recognition theorems
for these 3-manifolds as well as theoretical generalizations of these results
to higher dimensions are established. The methods developed build a bridge
linking geometric topology with iterated function systems and their attractors.
A method to model self-affine tiles by simple iterative systems is developed
in order to study their topology. The model is functorial in the sense that
there is an easily computable map that induces isomorphisms between the natural
subdivisions of the attractor of the model and the self-affine tile. It has
many beneficial qualities including ease of computation allowing one to
determine topological properties of the attractor of the model such as
connectedness and whether it is a manifold. The induced map between the
attractor of the model and the self-affine tile is a quotient map and can be
checked in certain cases to be monotone or cell-like. Deep theorems from
geometric topology are applied to characterize and develop algorithms to
recognize when a self-affine tile is a topological or generalized manifold in
all dimensions. These new tools are used to check that several self-affine
tiles in the literature are 3-balls. An example of a wild 3-dimensional
self-affine tile is given whose boundary is a topological 2-sphere but which is
not itself a 3-ball. The paper describes how any 3-dimensional handlebody can
be given the structure of a self-affine 3-manifold. It is conjectured that
every self-affine tile which is a manifold is a handlebody.Comment: 40 pages, 13 figures, 2 table
Spatially independent martingales, intersections, and applications
We define a class of random measures, spatially independent martingales,
which we view as a natural generalisation of the canonical random discrete set,
and which includes as special cases many variants of fractal percolation and
Poissonian cut-outs. We pair the random measures with deterministic families of
parametrised measures , and show that under some natural
checkable conditions, a.s. the total measure of the intersections is H\"older
continuous as a function of . This continuity phenomenon turns out to
underpin a large amount of geometric information about these measures, allowing
us to unify and substantially generalize a large number of existing results on
the geometry of random Cantor sets and measures, as well as obtaining many new
ones. Among other things, for large classes of random fractals we establish (a)
very strong versions of the Marstrand-Mattila projection and slicing results,
as well as dimension conservation, (b) slicing results with respect to
algebraic curves and self-similar sets, (c) smoothness of convolutions of
measures, including self-convolutions, and nonempty interior for sumsets, (d)
rapid Fourier decay. Among other applications, we obtain an answer to a
question of I. {\L}aba in connection to the restriction problem for fractal
measures.Comment: 96 pages, 5 figures. v4: The definition of the metric changed in
Section 8. Polishing notation and other small changes. All main results
unchange
The set of maps F_{a,b}: x -> x+a+{b/{2 pi}} sin(2 pi x) with any given rotation interval is contractible
Consider the two-parameter family of real analytic maps which are lifts of degree one endomorphisms of
the circle. The purpose of this paper is to provide a proof that for any closed
interval , the set of maps whose rotation interval is , form a
contractible set
Strong Non-Ultralocality of Ginsparg-Wilson Fermionic Actions
It is shown that it is impossible to construct a free theory of fermions on
infinite hypercubic Euclidean lattice in even number of dimensions that: (a) is
ultralocal, (b) respects the symmetries of hypercubic lattice, (c) chirally
nonsymmetric part of its propagator is local, and (d) describes single species
of massless Dirac fermions in the continuum limit. This establishes
non-ultralocality for arbitrary doubler-free Ginsparg-Wilson fermionic action
with hypercubic symmetries ("strong non-ultralocality"), and complements the
earlier general result on non-ultralocality of infinitesimal
Ginsparg-Wilson-Luscher symmetry transformations ("weak non-ultralocality").Comment: 21 pages, 1 figure, LATEX. Few typos corrected; few sentences
reformulated; figure centere
Triangular Gatzouras-Lalley-type planar carpets with overlaps
We construct a family of planar self-affine carpets with overlaps using lower
triangular matrices in a way that generalizes the original Gatzouras--Lalley
carpets defined by diagonal matrices. Assuming the rectangular open set
condition, Bara\'nski proved for this construction that for typical parameters,
which can be explicitly checked, the inequalities between the Hausdorff, box
and affinity dimension of the attractor are strict. We generalize this result
to overlapping constructions, where we allow complete columns to be shifted
along the horizontal axis or allow parallelograms to overlap within a column in
a transversal way. Our main result is to show sufficient conditions under which
these overlaps do not cause the drop of the dimension of the attractor. Several
examples are provided to illustrate the results, including a self-affine
smiley, a family of self-affine continuous curves, examples with overlaps and
an application of our results to some three-dimensional systems.Comment: 12 figures; v2: improved presentation, updated references, added a
three-dimensional example and an Appendix. Results unchange
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