相关论文: Fractal Measures, p-Adic Numbers And Continues Tra…

A class of simplified measures is constructed to capture the key features of generic spatio-temporally chaotic systems. A combined analytical and numerical investigation allows us to extablish the scaling beahviour of the fractal dimension…

A lot of formal and informal recreational study took place in the fields of Meromorphic Maps, since Mandelbrot popularized the map z <- z^2 + c. An immediate generalization of the Mandelbrot z <-z^n + c also known as the Multibrot family…

This paper investigates a class of deterministic fractals whose construction is governed by arithmetic sequences. We introduce the essential fractal prime set P_{ess} , a variant of the Cantor set constructed using the sequence of prime…

We study the Hausdorff and box-counting dimensions of cookie-cutter-like sets formed by sequential dynamics of a finite number of expanding maps. Under some natural conditions, these dimensions turn out to be the minimum and maximum of the…

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, in a product space whose alphabet is a countable set. More specifically, we show…

We introduce a novel deterministic fractal set PF in the unit interval whose construction is driven by the sequence of prime numbers modulo 16. At each step of the recursive construction, two subintervals are retained based on the residues…

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's…

Covariant (Lorentz invariant) fracton matter, minimally coupled and charged under a symmetric rank two gauge tensor, is considered. The gauge transformations correspond to linearized longitudinal diffeomorphisms. Consistent possible…

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…

In this paper, we prove the identity $\dim_{\textrm H}(F)=d\cdot \dim_{\textrm H}(\alpha^{-1}(F))$, where $\dim_{\textrm H}$ denotes Hausdorff dimension, $F\subseteq \mathbb{R}^d$, and $\alpha:[0,1]\to [0,1]^d$ is a function whose…

We study the dynamics of a family of continued fraction maps parametrized by the unit interval. This family contains as special instances the Gauss continued fraction map and the Fibonacci map. We determine the transfer operators of these…

Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections'' of a "network" corresponding to a fractal set, $F$. This lead to the definition of the…

Although geographic features, such as mountains and coastlines, are fractal, some studies have claimed that the fractal property is not universal. This claim, which is false, is mainly attributed to the strict definition of fractal…

We study various measure theories using the classical approach and then compute the Hausdorff dimension of some simple objects and self-similar fractals. We then develop a nonstandard approach to these measure theories and examine the…

We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and…

We determine the Hausdorff and box dimension of the fractal graphs for a general class of Weierstrass-type functions of the form $f(x) = \sum_{n=1}^\infty a_n \, g(b_n x + \theta_n)$, where $g$ is a periodic Lipschitz real function and…

A mathematical method for constructing fractal curves and surfaces, termed the $p\lambda n$ fractal decomposition, is presented. It allows any function to be split into a finite set of fractal discontinuous functions whose sum is equal…

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…

Antoniadis, Mazur and Mottola (AMM) two years ago computed the intrinsic Hausdorff dimension of spacetime at the infrared fixed point of the quantum conformal factor in 4D Gravity. The fractal dimension was determined by the coefficient of…

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$ for all…