6,662 research outputs found
Exact Hausdorff and packing measures of linear Cantor sets with overlaps
Let be the attractor of a linear iterated function system (IFS)
, on the real line satisfying the
generalized finite type condition (whose invariant open set is an
interval) with an irreducible weighted incidence matrix. This condition was
introduced by Lau \& Ngai recently as a natural generalization of the open set
condition, allowing us to include many important overlapping cases. They showed
that the Hausdorff and packing dimensions of coincide and can be calculated
in terms of the spectral radius of the weighted incidence matrix. Let
be the dimension of . In this paper, we state that \begin{equation*}
\mathcal{H}^\alpha(K\cap J)\leq |J|^\alpha \end{equation*} for all intervals
, and \begin{equation*}
\mathcal{P}^\alpha(K\cap J)\geq |J|^\alpha \end{equation*} for all intervals
centered in , where
denotes the -dimensional Hausdorff measure and
denotes the -dimensional packing measure. This result extends a recent
work of Olsen where the open set condition is required. We use these
inequalities to obtain some precise density theorems for the Hausdorff and
packing measures of . Moreover, using these densities theorems, we describe
a scheme for computing exactly as the minimum of a
finite set of elementary functions of the parameters of the IFS. We also obtain
an exact algorithm for computing as the maximum of
another finite set of elementary functions of the parameters of the IFS. These
results extend previous ones by Ayer \& Strichartz and Feng, respectively, and
apply to some new classes allowing us to include linear Cantor sets with
overlaps.Comment: 41 pages, 5 figure
Continuity of packing measure function of self-similar iterated function systems
In this paper, we focus on the packing measure of self-similar sets. Let
be a self-similar set whose Hausdorff dimension and packing dimension equal
, we state that if satisfies the strong open set condition with an open
set , then for each open
ball centered in , where denotes
the -dimensional packing measure. We use this inequality to obtain some
precise density theorems for packing measure of self-similar sets, which can be
applied to compute the exact value of the -dimensional packing measure
of . Moreover, by using the above results, we show the
continuity of the packing measure function of the attractors on the space of
self-similar iterated function systems satisfying the strong separation
condition. This result gives a complete answer to a question posed by L. Olsen.Comment: 17 page
Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions
We consider the Sobolev type spaces with , where
is a post-critically finite self-similar set with the natural boundary.
Firstly, we compare different classes of Sobolev spaces
and , {and observe} a sequence of
critical orders of in our comparison theorem. Secondly, We present a
general atomic decomposition theorem of Sobolev spaces , where the
same critical orders play an important role. At the same time, we provide
purely analytic approaches for various Besov type characterizations of Sobolev
spaces .Comment: 39 pages, 1 figure. This is an update of arXiv:1904.0034
Some Properties of the Derivatives on Sierpinski Gasket Type Fractals
In this paper, we focus on Strichartz's derivatives, a family of derivatives
including the normal derivative, on p.c.f. (post critically finite) fractals,
which are defined at vertex points in the graphs that approximate the fractal.
We obtain a weak continuity property of the derivatives for functions in the
domain of the Laplacian. For a function with zero normal derivative at any
fixed vertex, the derivatives, including the normal derivatives of the
neighboring vertices will decay to zero. The optimal rates of approximations
are described and several non-trivial examples are provided to illustrate that
our estimates are sharp. We also study the boundness property of derivatives
for functions in the domain of the Laplacian. A necessary condition for a
function having a weak tangent of order one at a vertex point is provided.
Furthermore, we give a counter-example of a conjecture of Strichartz on the
existence of higher order weak tangents.Comment: 25 pages, 7 figure
A topological proof of the non-degeneracy of harmonic structures on Sierpinski Gaskets
We present a direct proof of the non-degeneracy of the harmonic structures on
the level- Sierpinski gaskets for any , which was conjectured by
Hino in [H1,H2] and confirmed to be true by Tsougkas [T] very recently using
Tutte's spring theorem.Comment: 5 pages, 3 figure
Higher order tangents and Higher order Laplacians on Sierpinski Gasket Type Fractals
We study higher order tangents and higher order Laplacians on p.c.f.
self-similar sets with fully symmetric structures, such as or
symmetric fractals. Firstly, let be a vertex point in the graphs that
approximate the fractal, we prove that for any defined near , the higher
oder weak tangent of at , if exists, is the uniform limit of local
multiharmonic functions that agree with in some sense near . Secondly,
we prove that the higher order Laplacian on a fractal can be expressible as a
renormalized uniform limit of higher order graph Laplacians on the graphs that
approximate the fractal. The main technical tool is the theory of local
multiharmonic functions and local monomials analogous to . The
results in this paper are closely related to the theory of local Taylor
approximations, splines and entire analytic functions. Some of our results can
be extended to general p.c.f. fractals. In Appendix of the paper, we provide a
recursion algorithm for the exact calculations of the boundary values of the
monomials for or symmetric fractals, which is shorter and more direct
than the previous work on the Sierpinski gasket.Comment: 42 pages, 13 figures, 4 table
Restrictions of Laplacian eigenfunctions to edges in the Sierpinski gasket
In this paper, we study the restrictions of both the harmonic functions and
the eigenfunctions of the symmetric Laplacian to edges of pre-gaskets contained
in the Sierpinski gasket. For a harmonic function, its restriction to any edge
is either monotone or having a single extremum. For an eigenfunction, it may
have several local extrema along edges. We prove general criteria, involving
the values of any given function at the endpoints and midpoint of any edge, to
determine which case it should be, as well as the asymptotic behavior of the
restriction near the endpoints. Moreover, for eigenfunctions, we use spectral
decimation to calculate the exact number of the local extrema along any edge.
This confirm, in a more general situation, a conjecture of K. Dalrymple, R.S.
Strichartz and J.P. Vinson \cite{DSV} on the behavior of the restrictions to
edges of the basis Dirichlet eigenfunctions, suggested by the numerical data.Comment: 23 pages, 9 figure
Dirichlet forms on self-similar sets with overlaps
We study Dirichlet forms and Laplacians on self-similar sets with overlaps. A
notion of "finitely ramified of finite type() nested structure" for
self-similar sets is introduced. It allows us to reconstruct a class of
self-similar sets in a graph-directed manner by a modified setup of Mauldin and
Williams, which satisfies the property of finite ramification. This makes it
possible to extend the technique developed by Kigami for analysis on
self-similar sets to this more general framework. Some basic properties related
to nested structures are investigated. Several non-trivial examples
and their Dirichlet forms are provided.Comment: 38 pages, 29 figure
Mean value properties of harmonic functions on Sierpinski gasket type fractals
In this paper, we establish an analogue of the classical mean value property
for both the harmonic functions and some general functions in the domain of the
Laplacian on the Sierpinski gasket. Furthermore, we extend the result to some
other p.c.f. fractals with Dihedral-3 symmetry.Comment: 27 pages, 9 figures, to appear, JFA
Open set condition and pseudo Hausdorff measure of self-affine IFSs
Let be an real expanding matrix and be a finite
subset of with . The family of maps
is called a self-affine iterated
function system (self-affine IFS). The self-affine set is
the unique compact set determined by satisfying the
set-valued equation . The
number with , is the so-called
pseudo similarity dimension of . As shown by He and Lau, one can associate
with and any number a natural pseudo Hausdorff measure denoted by
In this paper, we show that, if is chosen to be the
pseudo similarity dimension of , then the condition
holds if and only if the IFS satisfies the open set
condition (OSC). This extends the well-known result for the self-similar case
that the OSC is equivalent to having positive Hausdorff measure
for a suitable . Furthermore, we relate the exact value of
pseudo Hausdorff measure to a notion of upper -density
with respect to the pseudo norm associated with for the measure
in the case that .Comment: 25page
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