64 research outputs found
Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems
The fast basin of an attractor of an iterated function system (IFS) is the
set of points in the domain of the IFS whose orbits under the associated
semigroup intersect the attractor. Fast basins can have non-integer dimension
and comprise a class of deterministic fractal sets. The relationship between
the basin and the fast basin of a point-fibred attractor is analyzed. To better
understand the topology and geometry of fast basins, and because of analogies
with analytic continuation, branched fractal manifolds are introduced. A
branched fractal manifold is a metric space constructed from the extended code
space of a point-fibred attractor, by identifying some addresses. Typically, a
branched fractal manifold is a union of a nondenumerable collection of
nonhomeomorphic objects, isometric copies of generalized fractal blowups of the
attractor
Symmetric Itinerary Sets
We consider a one parameter family of dynamical systems W :[0, 1] -> [0, 1]
constructed from a pair of monotone increasing diffeomorphisms Wsub(i), such
that Wsub(i)(inverse): [0, 1] -> [0, 1], (i = 0, 1). We characterise the set of
symbolic itineraries of W using an attractor of an iterated closed relation,in
the terminology of McGehee, and prove that there is a member of the family for
which is symmetrical
Bilinear Fractal Interpolation and Box Dimension
In the context of general iterated function systems (IFSs), we introduce
bilinear fractal interpolants as the fixed points of certain
Read-Bajraktarevi\'{c} operators. By exhibiting a generalized "taxi-cab"
metric, we show that the graph of a bilinear fractal interpolant is the
attractor of an underlying contractive bilinear IFS. We present an explicit
formula for the box-counting dimension of the graph of a bilinear fractal
interpolant in the case of equally spaced data points
Numerics and Fractals
Local iterated function systems are an important generalisation of the
standard (global) iterated function systems (IFSs). For a particular class of
mappings, their fixed points are the graphs of local fractal functions and
these functions themselves are known to be the fixed points of an associated
Read-Bajactarevi\'c operator. This paper establishes existence and properties
of local fractal functions and discusses how they are computed. In particular,
it is shown that piecewise polynomials are a special case of local fractal
functions. Finally, we develop a method to compute the components of a local
IFS from data or (partial differential) equations.Comment: version 2: minor updates and section 6.1 rewritten, arXiv admin note:
substantial text overlap with arXiv:1309.0243. text overlap with
arXiv:1309.024
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