Related papers: Spectral asymptotics for Stretched Fractals
The Hanoi attractor (or Stretched Sierpinski Gasket) is an example of a non-self-similar fractal that still exhibits a lot of symmetry. The existence of various symmetric resistance forms on the Hanoi attractor was shown in 2016 by…
The asymptotic behaviour of the eigenvalue counting function of Laplacians on Hanoi attractors is determined. To this end, Dirichlet and resistance forms are constructed. Due to the non self-similarity of these sets, the classical…
In this paper, we introduce the notion of asymptotic self-similar sets on general doubling metric spaces by extending the notion of self-similar sets, and determine their Hausdorff dimensions, which gives an extension of Balogh and Rohner…
The stretched Sierpinski gasket, SSG for short, is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. It has also been called "deformed Sierpinski gasket" or "Hanoi attractor". As a result, it is…
We estabish rigorous estimates for the Hausdorff dimension of the spectra of Laplacians associated to Sierpi\'nski lattices and infinite Sierpi\'nski gaskets and other post-critically finite self-similar sets.
Fractal geometry is the study of sets which exhibit the same pattern at multiple scales. Developing tools to study these sets is of great interest. One step towards developing some of these tools is recognizing the duality between…
The Stretched Sierpinski Gasket is a non-self-similar set but it still exhibits very high symmetry. In an earlier work we calculated the leading term for the eigenvalue counting function for operators coming from resistance forms that were…
In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study…
In the present paper we define statistically self-similar sets, and, using a modification of method described K.J.Falconer find a Hausdorff dimension of a statistically self-similar set.
We study the convergence of resistance metrics and resistance forms on a converging sequence of spaces. As an application, we study the existence and uniqueness of self-similar Dirichlet forms on Sierpinski gaskets with added rotated…
This article studies potential theory and spectral analysis on compact metric spaces, which we refer to as fractal quantum graphs. These spaces can be represented as a (possibly infinite) union of 1-dimensional intervals and a totally…
We first establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a point and the leading term of the asymptotic expansion of the Dirichlet eigenvalue variation, as a removed compact set concentrates at that…
This article deals with (1) the construction of a general non-linear fractal interpolation function on PCF self-similar sets, (2) the energy and normal derivatives of uniform non-linear fractal functions, (3) estimation of the bound of box…
We study spectral asymptotics for small non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori…
In this paper, we study discrete approximations of semi-Dirichlet forms obtained by adding non-symmetric drift terms, expressed in terms of mutual energy measures, to resistance forms whose associated resistance metric spaces are compact.…
We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application we determine the mean and…
We construct a family of spectral triples for the Sierpinski Gasket $K$. For suitable values of the parameters, we determine the dimensional spectrum and recover the Hausdorff measure of $K$ in terms of the residue of the volume functional…
We present a new approach to the theory of k-forms on self-similar fractals. We work out the details for two examples, the standard Sierpinski gasket and the 3-dimensional Sierpinski gasket, but the method is expected to be effective for…
In the paper, we define a class of new fractals named ``non-autonomous attractors", which are the generalization of classic Moran sets and attractors of iterated function systems. Simply to say, we replace the similarity mappings by…
Let $\Gamma_n$ denote the $n$-th level Sierpi\'nski graph of the Sierpi\'nski gasket $K$. We consider, for any given conductance $(a_0, b_0, c_0)$ on $\Gamma_0$, the Dirchlet form ${\mathcal E}$ on $K$ obtained from a recursive construction…