相关论文: Fractals in Noncommutative Geometry

Two spectral triples are introduced for a class of fractals in R^n. The definitions of noncommutative Hausdorff dimension and noncommutative tangential dimensions, as well as the corresponding Hausdorff and Hausdorff-Besicovitch functionals…

Given a spectral triple (A,D,H), the functionals on A of the form a -> tau_omega(a|D|^(-t)) are studied, where tau_omega is a singular trace, and omega is a generalised limit. When tau_omega is the Dixmier trace, the unique exponent d…

This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a…

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…

We compute the intrinsic Hausdorff dimension of spacetime at the infrared fixed point of the quantum conformal factor in 4D gravity. The fractal dimension is defined by the appropriate covariant diffusion equation in four dimensions and is…

It is shown that, for nested fractals [T.Lindstrom, Mem. Amer. Math. Soc. 420, 1990], the main structural data, such as the Hausdorff dimension and measure, the geodesic distance (when it exists) induced by the immersion in $R^n$, and the…

There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $\mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $\mathbb Z^d$ and, more generally, for…

Fractal measures of images of continuous maps from the set of p-adic numbers Qp into complex plane C are analyzed. Examples of "anomalous" fractals, i.e. the sets where the D-dimensional Hausdorff measures (HM) are trivial, i.e. either…

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space.…

Singular vectors are those for which the quality of rational approximations provided by Dirichlet's Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular…

We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain $\mathbb{R}^n$ is assumed to be the compact attractor of an iterated function system of…

We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…

We show how multifractal properties of a measure supported by a fractal F contained in [0,1] may be expressed in terms of complementary intervals of F and thus in terms of spectral triples and the Dixmier trace of certain operators. For…

We calculate the almost sure Hausdorff dimension of uniformly random self-similar fractals. These random fractals are generated from a finite family of similarities, where the linear parts of the mappings are independent uniformly…

This paper presents a simple method of calculating the Hausdorff dimension for a class of non-conformal fractals.

Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure:…

Two examples of spectral triples with non-integer dimension spectrum are considered. These triples involve commutative C*-algebras. The first example has complex dimension spectrum and trivial differential algebra. The other is a parameter…

We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets,…

Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff…

Fractal Lipschitz-Killing curvature measures C^f_k(F,.), k = 0, ..., d, are determined for a large class of self-similar sets F in R^d. They arise as weak limits of the appropriately rescaled classical Lipschitz-Killing curvature measures…