Related papers: Spectral triples for the Sierpinski Gasket

We introduce a Lindblad-deformed spectral geometric framework in which bounded dissipative data deform a standard spectral triple through the Dirac operator D_gamma = D - igammaSigma, where Sigma = (1/2) sum_k L_k^dagger L_k is constructed…

We show that the family of spectral triples for quantum projective spaces introduced by D'Andrea and Dabrowski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action…

The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance…

Following \cite{Visintin}, we exploit the fractional perimeter of a set to give a definition of fractal dimension for its measure theoretic boundary. We calculate the fractal dimension of sets which can be defined in a recursive way and we…

We present measurements of the fractal dimension of a turbulent asymptotically anti-deSitter black brane reconstructed from simulated boundary fluid data at the perfect fluid order using the fluid-gravity duality. We argue that the boundary…

The family of $V$-variable fractals provides a means of interpolating between two families of random fractals previously considered in the literature; scale irregular fractals ($V=1$) and random recursive fractals ($V=\infty$). We consider…

We explore the relation between noncommutative geometry, in the spectral triple formulation, and quantum mechanics. To this aim, we consider a dynamical theory of a noncommutative geometry defined by a spectral triple, and study its…

We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for…

We explicitly construct Fredholm modules and spectral triples representing any element of $K$-homology groups of Hensel-Steinitz algebras.

J. Kigami has laid the foundations of what is now known as analysis on fractals, by allowing the construction of an operator of the same nature of the Laplacian, defined locally, on graphs having a fractal character. The Sierpinski gasket…

Let $S \subset {\mathbb R}^d$ be contained in the unit ball. Let $\Delta(S)=\{||a-b||:a,b \in S\}$, the Euclidean distance set of $S$. Falconer conjectured that the $\Delta(S)$ has positive Lebesque measure if the Hausdorff dimension of $S$…

As an emerging complex two-dimensional structure, plane fractal has attracted much attention due to its novel dimension-related physical properties. In this paper, we check the feasibility to create an effective Sierpinski carpet (SC), a…

We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge $K\subset \mathbb{R}^k$ ($k\ge 2$) obtained by using some percolation process in $[0,1]^k$. To do so, we first exhibit a Ledrappier-Young type…

Quasisymmetric maps are well-studied homeomorphisms between metric spaces preserving annuli, and the Ahlfors regular conformal dimension $\dim_\mathrm{ARC}(X,d)$ of a metric space $(X,d)$ is the infimum over the Hausdorff dimensions of the…

We propose a construction for spectral triple on algebras associated with subshifts. One-dimensional subshifts provide concrete examples Z-actions on Cantor sets. The C*-algebra of this dynamical system is generated by functions in C(X) and…

We prove that if ${\mathcal E} \subset {\Bbb R}^{2d}$, $d \ge 2$, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by $dim_{{\mathcal H}}({\mathcal E})$, and $\phi$ is a sufficiently regular…

We prove that the infinitely generated Apollonian gasket has full Hausdorff dimension spectrum. Our proof, which is computer assisted, relies on an iterative technique introduced by the first three authors in [3] and on a flexible method…

We consider the Hausdorff dimension of the divergence set on which the pointwise convergence $\lim_{t\rightarrow 0} e^{it\sqrt{-\Delta}} f(x) = f(x)$ fails when $f \in H^s(\mathbb R^d)$. We especially prove the conjecture raised by…

In this paper we show that the Hausdorff dimension of the set of singular pairs is 4/3. We also show that the action of diag(e^t,e^t,e^{-2t}) on SL(3,R)/SL(3,Z) admits divergent trajectories that exit to infinity at arbitrarily slow…

We introduce a model of interacting bosons exhibiting an infinite collection of fractal symmetries -- termed "Pascal's triangle symmetries" -- which provides a natural $U(1)$ generalization of a spin-(1/2) system with Sierpinski triangle…