ABSOLUTE CONTINUITY FOR RANDOM ITERATED FUNCTION SYSTEMS WITH OVERLAPS

The eigenvalue problem for linear and affine iterated function systems

We study parabolic iterated function systems (IFS) with overlaps on the real line and measures associated with them. A Borel probability measure μ on the coding space projects into a measure v on the limit set of the IFS.We consider families of IFS satisfying a transversality condition. In [SSU2] sufficient conditions were found for the measure v to be absolutely continuous for Lebesgue-a.e. parameter value. Here we investigate when v has a density in L q (R) for q > 1. A necessary condition is that the q-dimension of μ (computed with respect to a certain metric associated with the IFS) is greater or equal to one. We prove that this is sharp for 1 < q < 2 in the following sense: if μ is a Gibbs measure with a Holder continuous potential, then v has a density in L q (R) for Lebesgue-a.e. parameter value such that the q-dimension of μ is greater than one. This result is applied to a family of random continued fractions studied by R. Lyons.

We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.

Chaotic dynamic characteristics in swarm intelligence

Let y = {ii, ... ,4t} be a finite set of coprime integers and let {l, 2,...} denote the mutiplicative semigroup generated by y, and arranged in increasing order.Let Dn(w) denote the discrepancy of the sequence {nkco}%=i mod 1, w e [0, 1).In this paper we solve a problem posed by R.C. Baker [3], by proving that for all w except on a set of Lebesgue measure 0 l<limsuP /*(ft,) <C.Here the constant C only depends on the total number of primes involved in the prime factorization of q ,... , qx.The lower bound is obtained from a strong approximation theorem for the partial sums of the sequence {cos2nnk<}kx>=l by sums of independent standard normal random variables.P{co : t]k((o) < x} = x, 0<JC<1, *>1.If Fff denotes the empirical distribution function of the sequence {nk} at stage N, i.e., FN(s) = FN(s ; to) := N-l*{k < N : nk(w) <s}, 0 < s < 1, then DN(co):= sup \Fjf(s) -s\ 0<s<X denotes the discrepancy (or the Kolmogorov-Smirnov statistic) of the sequence {jt<y}f=1 mod 1. (For the basic definition and facts on uniform distribution

Approximation of an image by the attractor evolved through iterations of a set of contractive maps is usually known as fractal image compression. The set of maps is called iterated function system (IFS). Several algorithms, with different motivations, have been suggested towards the solution of this problem. But, so far, the theory of IFS with probabilities, in the context of image compression, has not been explored much. In the present article we have proposed a new technique of fractal image compression using the theory of IFS and probabilities. In our proposed algorithm, we have used a multiscaling division of the given image up to a predetermined level or up to that level at which no further division is required. At each level, the maps and the corresponding probabilities are computed using the gray value information contained in that image level and in the image level higher to that level. A fine tuning of the algorithm is still to be done. But, the most interesting part of the proposed technique is its extreme fastness in image encoding. It can be looked upon as one of the solutions to the problem of huge computational cost for obtaining fractal code of images.

The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Techniques from probabilistic number theory are used to establish necessary and sufficient conditions for weak convergence of the associated dependent process to a process with independent increments. Not many results on the necessity part are known in the literature.

On some inequalities in the probabilistic number theory

On the class of labelled combinatorial structures called assemblies we define complex-valued multiplicative functions and examine their asymptotic mean values. The problem reduces to the investigation of quotients of the Taylor coefficients of exponential generating series having Euler products. Our approach, originating in probabilistic number theory, requires information on the generating functions only in the convergence disc and rather weak smoothness on the circumference. The results could be applied to studying the asymptotic value distribution of decomposable mappings defined on assemblies.

Sums of multiplicative functions and limit theorems in probabilistic number theory. I

ABSTRACTThe aim of this paper is to give some new characterizations of discrete compound Poisson distributions. Firstly, we give a characterization by the Lévy–Khintchine formula of infinitely divisible distributions under some conditions. The second characterization need to present by row sum of random triangular arrays converges in distribution. And we give an application in probabilistic number theory, the strongly additive function converging to a discrete compound Poisson in distribution. The next characterization, is an extension of Watanabe’s theorem of characterization of homogeneous Poisson process. The last characterization will be illustrated by waiting time distributions, especially the matrix-exponential representation.

The paper presents an overview of main methods and results of the value distribution problem of additive arithmetical functions.

Size biased sampling from the Dickman subordinator

Let (nk)k≥1 be an increasing sequence of positive integers, and let f (x) be a real function satisfyingIf the distribution ofconverges to a Gaussian distribution. In the casethere is a complex interplay between the analytic properties of the function f , the number-theoretic properties of (nk)k≥1, and the limit distribution of (2).In this paper we prove that any sequence (nk)k≥1 satisfying lim contains a nontrivial subsequence (mk)k≥1 such that for any function satisfying (1) the distribution ofconverges to a Gaussian distribution. This result is best possible: for any ε &gt; 0 there exists a sequence (nk)k≥1 satisfying lim such that for every nontrivial subsequence (mk)k≥1 of (nk)k≥1 the distribution of (2) does not converge to a Gaussian distribution for some f.Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

We treat synchronization for iterated function systems generated by diffeomorphisms on compact manifolds. Synchronization here means the convergence of orbits starting at different initial conditions when iterated by the same sequence of diffeomorphisms. The iterated function systems admit a description as skew product systems of diffeomorphisms on compact manifolds driven by shift operators. Under open conditions including transitivity and negative fiber Lyapunov exponents, we prove the existence of a unique attracting invariant graph for the skew product system. This explains the occurrence of synchronization. The result extends previous results for iterated function systems by diffeomorphisms on the circle, to arbitrary compact manifolds.