Related papers: Dimension of invariant measures for affine iterate…

We prove that if $J$ is the limit set of an irreducible conformal iterated function system (with either finite or countably infinite alphabet), then the badly approximable vectors form a set of full Hausdorff dimension in $J$. The same is…

For a Borel probability measure $\mu$ on $\mathbb{R}^{n}$, it is called a spectral measure if the Hilbert space $L^{2}(\mu)$ admits an orthogonal basis of exponential functions. In this paper, we study the spectrality of fractal measures…

In the paper, we define a class of new fractals named ``non-autonomous attractors", which are the generalization of classic Moran sets and attractors of iterated function systems. Simply to say, we replace the similarity mappings by…

Assuming strong irreducibility and proximality, we prove that the Furstenberg measure, corresponding to a finitely supported measure on the general linear group of a finite dimensional real vector space, is exact dimensional. We also…

Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the…

The attractors of iterated function systems are usually obtained as the Hausdorff limit of any non-empty compact subset under iteration. In this note we show that an iterated function system on a boundedly compact metric space has compact,…

We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\bf R$ and other more general IFS's. We extend some of the theory recently developed for Bernoulli…

We consider a family S=S(a) of 2-valued transformations of special form on the segment [0,1] with measure $\mu=\int p(x) d\lambda$, which is absolutely continuous with respect to the Lebesgue measure $\lambda$. We endow S with a set of…

Shrinking target problems in the context of iterated function systems have received an increasing amount of interest in the past few years. The classical shrinking target problem concerns points returning infinitely many times to a sequence…

In 1988 K. Falconer introduced a formula which predicts the value of the Hausdorff dimension of the attractor of an affine iterated function system. The value given by this formula -- sometimes referred to as the affinity dimension -- is…

The method of iterated conformal maps allows to study the harmonic measure of Diffusion Limited Aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of…

The exact joint multifractal distribution for the scaling and winding of the electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff dimension…

K. Igusa and G. Todorov introduced two functions $\phi$ and $\psi,$ which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become into a powerful tool to…

As a natural counterpart to Nakada's $\alpha$-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter…

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…

The orbit of a point $x\in X$ in a classical iterated function system (IFS) can be defined as $\{f_u(x)=f_{u_n}\circ\cdots \circ f_{u_1}(x):$ $u=u_1\cdots u_n$ is a word of a full shift $\Sigma$ on finite symbols and $f_{u_i}$ is a…

We consider iterated function systems on the interval with random perturbation. Let $Y_\epsilon$ be uniformly distributed in $[1- \epsilon, 1 + \epsilon]$ and let $f_i \in C^{1+\alpha}$ be contractions with fixpoints $a_i$. We consider the…

Measures generated by Iterated Function Systems composed of uncountably many one--dimensional affine maps are studied. We present numerical techniques as well as rigorous results that establish whether these measures are absolutely or…

We consider the family of CIFSs of generalized complex continued fractions with a complex parameter space. This is a new interesting example to which we can apply a general theory of infinite CIFSs and analytic families of infinite CIFSs.…

Uniformly perfect measures are a common generalisation of Ahlfors regular measures, self-conformal measures on the line, and their push-forwards under sufficiently regular maps. We show that every uniformly perfect measure $\sigma$ on a…