Exact Hausdorff and packing measures of linear Cantor sets with overlaps

Let $K$ be the attractor of a linear iterated function system (IFS) $S_j(x)=\rho_jx+b_j,$ $j=1,\cdots,m$, on the real line satisfying the generalized finite type condition (whose invariant open set $\mathcal{O}$ 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 $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let $\alpha$ be the dimension of $K$. In this paper, we state that \begin{equation*} \mathcal{H}^\alpha(K\cap J)\leq |J|^\alpha \end{equation*} for all intervals $J\subset\overline{\mathcal{O}}$, and \begin{equation*} \mathcal{P}^\alpha(K\cap J)\geq |J|^\alpha \end{equation*} for all intervals $J\subset\overline{\mathcal{O}}$ centered in $K$, where $\mathcal{H}^\alpha$ denotes the $\alpha$-dimensional Hausdorff measure and $\mathcal{P}^\alpha$ denotes the $\alpha$-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 $K$. Moreover, using these densities theorems, we describe a scheme for computing $\mathcal{H}^\alpha(K)$ 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 $\mathcal{P}^\alpha(K)$ 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 $K$ be a self-similar set whose Hausdorff dimension and packing dimension equal $s$, we state that if $K$ satisfies the strong open set condition with an open set $\mathcal{O}$, then $\mathcal{P}^s(K\cap B(x,r))\geq (2r)^s$ for each open ball $B(x,r)\subset \mathcal{O}$ centered in $K$, where $\mathcal{P}^s$ denotes the $s$-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 $s$-dimensional packing measure $\mathcal{P}^s(K)$ of $K$. 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 $H^\sigma(K)$ with $\sigma\geq 0$, where $K$ is a post-critically finite self-similar set with the natural boundary. Firstly, we compare different classes of Sobolev spaces $H^\sigma_N(K),H^\sigma_D(K)$ and $H^\sigma(K)$, {and observe} a sequence of critical orders of $\sigma$ in our comparison theorem. Secondly, We present a general atomic decomposition theorem of Sobolev spaces $H^\sigma(K)$, 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 $H^\sigma(K)$.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-$n$ Sierpinski gaskets for any $n\geq 2$, 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 $D3$ or $D4$ symmetric fractals. Firstly, let $x$ be a vertex point in the graphs that approximate the fractal, we prove that for any $f$ defined near $x$, the higher oder weak tangent of $f$ at $x$, if exists, is the uniform limit of local multiharmonic functions that agree with $f$ in some sense near $x$. 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 $(x-x_0)^j/j!$. 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 $D3$ or $D4$ 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($f.r.f.t.$) 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 $p.c.f.$ self-similar sets to this more general framework. Some basic properties related to $f.r.f.t.$ 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 $A$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$ with $0\in\mathcal{D}$. The family of maps $\{f_d(x)=A^{-1}(x+d)\}_{d\in\mathcal{D}}$ is called a self-affine iterated function system (self-affine IFS). The self-affine set $K=K(A,\mathcal{D})$ is the unique compact set determined by $(A, {\mathcal D})$ satisfying the set-valued equation $K=\displaystyle\bigcup_{d\in\mathcal{D}}f_d(K)$. The number $s=n\,\ln(\# \mathcal{D})/\ln(q)$ with $q=|\det(A)|$, is the so-called pseudo similarity dimension of $K$. As shown by He and Lau, one can associate with $A$ and any number $s\ge 0$ a natural pseudo Hausdorff measure denoted by $\mathcal{H}_w^s.$ In this paper, we show that, if $s$ is chosen to be the pseudo similarity dimension of $K$, then the condition $\mathcal{H}_w^s(K)> 0$ holds if and only if the IFS $\{f_d\}_{d\in\mathcal{D}}$ satisfies the open set condition (OSC). This extends the well-known result for the self-similar case that the OSC is equivalent to $K$ having positive Hausdorff measure $\mathcal{H}^s$ for a suitable $s$. Furthermore, we relate the exact value of pseudo Hausdorff measure $\mathcal{H}_w^s(K)$ to a notion of upper $s$-density with respect to the pseudo norm $w(x)$ associated with $A$ for the measure $\mu=\lim\limits_{M\to\infty}\sum\limits_{d_0,\dotsc,d_{M-1}\in\mathcal{D}}\delta_{d_0 + Ad_1 + \dotsb + A^{M-1}d_{M-1}}$ in the case that $\#\mathcal{D}\le\lvert\det A\rvert$.Comment: 25page