Related papers: Complex dimensions for IFS with overlaps
A theory of resource-bounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound \Delta (a parameter of the theory) is unrestricted, the resulting dimension is precisely the…
This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the…
We present an overview of a theory of complex dimensions of self-similar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several…
The dynamics of a nonequilibrium system can become complex because the system has many components (e.g., a human brain), because the system is strongly driven from equilibrium (e.g., large Reynolds-number flows), or because the system…
In a previous paper, dealing with "Applications in $\mathbb{R}^1$," the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS and studied some applications…
Let f be a unimodal map of the interval with critical point c. If the orbit of c is not dense then most points in lim{[0,1],f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without…
We give new arguments for sums and products of sufficient numbers of arbitrary central Cantor sets to produce large open intervals. We further discuss the same question for $C^1$ images of such central Cantor sets. This gives another…
By using the generalized Bernoulli numbers, we deduce new integral representations for the Riemann zeta function at positive odd-integer arguments. The explicit expressions enable us to obtain criteria for the dimension of the vector space…
In this paper we consider the long-term behavior of points in ${\mathbb R}$ under iterations of continuous functions. We show that, given any Cantor set $\Lambda^*$ embedded in ${\mathbb R}$, there exists a continuous function $F^*:{\mathbb…
Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure $\mu$ vanishes, then the set of points where the principal value of the Cauchy singular integral of $\mu$ exists has…
Using an iterative tree construction we show that for simple computable subsets of the Cantor space Hausdorff, constructive and computable dimensions might be incomputable.
We construct a time-dependent expression of the computational complexity of a quantum system which consists of two conformal complex scalar field theories in d dimensions coupled to constant electric potentials and defined on the boundaries…
It has recently been realized that fractals may be characterized by complex dimensions, arising from complex poles of the corresponding zeta function, and we show here that these lead to oscillatory behavior in various physical quantities.…
We show how geometric methods from the general theory of fractal dimensions and iterated function systems can be deployed to study symbolic dynamics in the zero entropy regime. More precisely, we establish a dimensional characterization of…
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…
Suppose that $\mathcal{C}$ is the space of all middle Cantor sets. We characterize all triples $(\alpha,~\beta,~\lambda)\in \mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$ that satisfy $C_\alpha- \lambda C_\beta=[-\lambda,~1]. $ Also all…
In this paper we define a new class of metric spaces, called multi-model Cantor sets. We compute the Hausdorff dimension and show that the Hausdorff measure of a multi-model Cantor set is finite and non-zero. We then show that a bilipschitz…
Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${\mathbb R}^N$, for any integer $N\ge1$. It is defined by…
We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure…
While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the…