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 zero, or sigma-infinite (D is the Hausdorff dimension (HD) of this set) are presented. Using the Caratheodory construction, the generalized scale-covariant HM (GHM) being non-trivial on such fractals are constructed. In particular, we present an example of 0-fractal, the continuum with HD=0 and nontrivial GHM invariant w.r.t. the group of all diffeomorphisms C. For conformal transformations of domains in R^n, the formula for the change of variables for GHM is obtained. The family of continuous maps Qp in C continuously dependent on "complex dimension" d in C is obtained. This family is such that: 1) if d=2(1), then the image of Qp is C (real axis in C); 2) the fractal measures coincide with the images of the Haar measure in Qp, and at d=2(1) they also coincide with the flat (linear) Lebesgue measure; 3) integrals of entire functions over the fractal measures of images for any compact set in Qp are holomorphic in d, similarly to the dimensional regularization method in QFT.
Results concerning the statists of $r$-adic processes and their fractal properties are reviewed. The connection between singular eigenstates of the statistical evolution of such processes and popular fractal curves is emphasized.
Explicit formulas are obtained for a family of continuous mappings of p-adic numbers $\Qp$ and solenoids $\Tp$ into the complex plane $\sC$ and the space \~$\Rs ^{3}$, respectively. Accordingly, this family includes the mappings for which the Cantor set and the Sierpinski triangle are images of the unit balls in $\Qn{2}$ and $\Qn{3}$. In each of the families, the subset of the embeddings is found. For these embeddings, the Hausdorff dimensions are calculated and it is shown that the fractal measure on the image of $\Qp$ coincides with the Haar measure on $\Qp$. It is proved that under certain …
We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but satisfies some expansion properties for the adjoint action. Using these dynamical results, we study diophantine properties of typical points on some self-similar fractals in R^d. As examples, we show that for any self-similar fractal K in R^d satisfying the open set condition (for instance any translate or dilate of Cantor's middle thirds set or of a Koch snowfl…
We construct first a class of Moran fractals in R^d with countably many generators and non-stationary contraction rates; at each step n, the contractions depend on n-truncated sequences, and are related to asymptotic letter frequencies. In some cases the sets of contractions may be infinite at each step. We show that the Hausdorff dimension of such a fractal is equal to the zero h of a pressure function. We prove that the dimensions of these sets depend real analytically on the frequencies. Next, we apply the above construction to obtain non-stationary fractals E(x; f) \subset R^d, associated …
In this paper, we continue an investigation of the $p$-adic Ising-Vannimenus model on the Cayley tree of an arbitrary order $k$ $(k\geq 2$). We prove the existence of $p$-adic quasi Gibbs measures by analyzing fixed points of multi-dimensional $p$-adic system of equations. We are also able to show the uniqueness of translation-invariant $p$-adic Gibbs measure. Finally, it is established the existence of the phase transition for the Ising-Vannimenus model depending on the order $k$ of the Cayley tree and the prime $p$. Note that the methods used in the paper are not valid in the real setting, s…
We compute explicitly the density of the invariant measure for the Reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For some other algorithms, which are known to have a unique invariant measure absolutely continuous with respect to Lebesgue measure, the invariant domain found by this method seems to have a fractal boundary,…
This paper continues the study of the structure of finite intersections of general multiplicative translates $\mathcal{C}(M_1,\ldots,M_n)=\frac{1}{M_1}Σ_{3,\bar{2}}\cap\cdots\cap\frac{1}{M_n}Σ_{3,\bar{2}}$ for integers $1\leq M_1<\cdots<M_n$, where $Σ_{3,\bar{2}}$ denotes the $3$-adic Cantor set. This study was motivated by questions concerning the discrete dynamical system on the $3$-adic integers $\mathbb{Z}_3$ given by multiplication by $2$. The exceptional set $\mathcal{E}(\mathbb{Z}_3)$ is defined to be the set of all elements of $\mathbb{Z}_3$ whose forward orbits under this action inter…
By appropriate choices of elements in the underlying iterated function system, methodology of fractal interpolation entitles one to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as $α$-fractal operator on $\mathcal{C}(I)$, the space of all real-valued continuous functions defined on a compact interval $I$. With an eye towards connecting fractal functions with other branches of mathematics, in this article, we continue to investigate fractal operator in more gen…
Due to its fractal nature, much about the area of the Mandelbrot set $M$ remains to be understood. While a series formula has been derived by Ewing and Schober to calculate the area of $M$ by considering its complement inside the Riemann sphere, to date the exact value of this area remains unknown. This paper presents new improved upper bounds for the area based on a parallel computing algorithm and for the 2-adic valuation of the series coefficients in terms of the sum-of-digits function.
In this paper, we study three aspects of the $p-$adic Möbius maps. One is the group $\mathrm{PSL}(2,\mathcal{O}_{p})$, another is the geometrical characterization of the $p-$adic Möbius maps and its application, and the other is different norms of the $p-$adic Möbius maps. Firstly, we give a series of equations of the $p-$adic Möbius maps in $\mathrm{PSL}(2,\mathcal{O}_{p})$ between matrix, chordal, hyperbolic and unitary aspects. Furthermore, the properties of $\mathrm{PSL}(2,\mathcal{O}_{p})$ can be applied to study the geometrical characterization, the norms, the decomposition theorem of $p…
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 topological Hausdorff dimension of fractals. In this paper we continue our study of the level sets of generic $1$-Hölder-$α$ functions. While in a previous paper we gave the initial definitions and established some properties of these generic level sets, in this paper we provide numerical estimates in the case of the Sierpiński triangle. These calculations g…
We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set $\{1,2\}^\mathbb{N}$ of directive sequences. For a given set $\mathcal{C}$ of two substitutions, we show that there exists a $\mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally indep…
Let $α\in(0,1)$ be irrational and $[0;a_1,a_2,\cdots]$ be the continued fraction expansion of $α$. Let $H_{α,V}$ be the Sturm Hamiltonian with frequency $α$ and coupling $V$, $Σ_{α,V}$ be the spectrum of $H_{α,V}$. The fractal dimensions of the spectrum have been determined by Fan, Liu and Wen (Erg. Th. Dyn. Sys.,2011) when $\{a_n\}_{n\ge1}$ is bounded. The present paper will treat the most difficult case, i.e, $\{a_n\}_{n\ge1}$ is unbounded. We prove that for $V\ge24$, $$ \dim_H\ Σ_{α,V}=s_*(V)\ \ \ \text{and}\ \ \ \bar{\dim}_B\ Σ_{α,V}=s^*(V), $$ where $s_*(V)$ and $s^*(V)$ are lower and upp…
