Hausdorff dimension of the limit set of conformal iterated function systems with overlaps

Hausdorff Dimension of the Limit Set of Conformal Iterated Function Systems with Overlaps

We give a new approach to the study of conformal iterated function systems with arbitrary overlaps. We provide lower and upper estimates for the Hausdorff dimension of the limit sets of such systems; these are expressed in terms of the topological pressure and the function d, counting overlaps. In the case when the function d is constant, we get an exact formula for the Hausdorff dimension. We ...

Hausdorff Dimension of Limit Sets for Parabolic Ifs with Overlaps

We study parabolic iterated function systems with overlaps on the real line. We show that if a d-parameter family of such systems satisfies a transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of 1 and the least zero of the pressure function. Moreover, the local dimension of the exceptional set of parameters is estimated. If t...

Quantization Dimension for Conformal Iterated Function Systems

On the Geometric Structure of the Limit Set of Conformal Iterated Function Systems

We consider infinite conformal function systems on Rd. We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some l-dimensional C1-submanifold with positive Hausdorff t-dimensional measure, where 0 < l < d and t is the Hausdorff dimension of the limit set. We then show that the closure of the limit set belongs to some l-dimensional affine subspace ...

Hausdorff Dimension of Random Fractals with Overlaps

The term fractal was first introduced by Mandelbrot to denote sets with highly irregular structures. Mandelbrot and others have then used fractals to model various natural phenomena (for example, [19, 2]). A mathematical development of fractals via the theory of (non-random) self-similar sets and measures was given by Hutchinson [12]. A family of contractive maps {Sj}^ on E m is called an itera...