Strong Open Set Condition Research Articles

Packing Measure and Dimension of Random Fractals

We consider random fractals generated by random recursive constructions. We prove that the box-counting and packing dimensions of these random fractals, K, equals α, their almost sure Hausdorff dimension. We show that some “almost deterministic” conditions known to ensure that the Hausdorff measure satisfies \(0 < H^\alpha (K) < \infty \) also imply that the packing measure satisfies 0< \(0 < P^\alpha (K) < \infty \). When these conditions are not satisfied, it is known \(0 = H^\alpha (K)\). Correspondingly, we show that in this case \(P^\alpha (K) = \infty \), provided a random strong open set condition is satisfied. We also find gauge functions φ(t) so that the \(P^\Phi \)-packing measure is finite.

Quantization dimension for conformal iterated function systems

The quantization dimension function is determined for a certain class of probability measures generated bya finite conformal iterated function system satisfying the strong open set condition.

Iterated Function System and Ruelle Operator

We consider a generalized iterated function system where the weights are variable functions. By using the Ruelle operator and a dynamical system consideration we prove that if the system is contractive and the weights are strictly positive functions and satisfy the Dini condition, then there exists a unique eigenmeasure (corresponding to the Ruelle operator) on the attractor. If in addition the maps are conformal and satisfy the open set condition, then we prove that they satisfy the strong open set condition, and by using this we can give a description of theLp-scaling spectrum and the multifractal structure of the eigenmeasure. The work extends some results of [Proc. London Math. Soc.73(1996), 105–154;Adv. Appl. Math.19(1997), 486–513;J. Statist. Phys.86(1997), 233–275;Indiana Univ. Math. J.42(1993), 367–411].

A fractal dimension estimate for a graph-directed IFS of non-similarities

Suppose a graph-directed iterated function system consists of maps f_e with upper estimates of the form d(f_e(x),f_e(y)) <= r_e d(x,y). Then the fractal dimension of the attractor K_v of the IFS is bounded above by the dimension associated to the Mauldin--Williams graph with ratios r_e. Suppose the maps f_e also have lower estimates of the form d(f_e(x),f_e(y)) >= r'_e d(x,y) and that the IFS also satisfies the strong open set condition. Then the fractal dimension of the attractor K_v of the IFS is bounded below by the dimension associated to the Mauldin--Williams graph with ratios r'_e. When r_e = r'_e, then the maps are similarities and this reduces to the dimension computation of Mauldin & Williams for that case.

Dynamical boundary of a self-similar set

Given a self-similar set E generated by a finite system Ψ of contracting similitudes of a complete metric space X we analyze a separation condition for Ψ , which is obtained if, in the open set condition, the open subset of X is replaced with an open set in the topology of E as a metric subspace of X. We prove that such a condition, which we call the restricted open set condition, is equivalent to the strong open set condition. Using the dynamical properties of the forward shift, we find a canonical construction for the largest open set V satisfying the restricted open set condition. We show that the boundary of V in E, which we call the dynamical boundary of E, is made up of exceptional points from a topological and measure-theoretic point of view, and it exhibits some other boundary-like properties. Using properties of subselfsimilar sets, we find a method which allows us to obtain the Hausdorff and packing dimensions of the dynamical boundary and the overlapping set in the case when X is the n-dimensional Euclidean space and Ψ satisfies the open set condition. We show that, in this case, the dimension of these sets is strictly less than the dimension of the set E.

On Hausdorff dimension of Julia sets of hyperbolic rational semigroups

We will show that if a semigroup of rational functions on the Riemann sphere is finitely generated, then the hyperbolicity and the expandingness are equivalent. Also we consider finitely generated rational semigroups satisfying the strong open set condition. We show that if a semigroup satisfies the strong open set condition, we can construct a ^-conformal measure on the Julia set. Also the Julia set has no interior points, and furthurmore, if the semigroup is hyperbolic, the Hausdorff dimension of the Julia set is strictly lower than 2. The value δ of the dimension coincides with the unique value that allows us to construct a <5-conformal measure and the <5-Hausdorff measure of the Julia set is a finite value strictly bigger than zero. With the method similar to that of the construction of the Patterson-Sullivan measures we get (5-subconformal measures in more general cases and we will show that if a finitely generated rational semigroup is expanding, then the HausdorfF dimension of the Julia set is less than the exponent δ.

The strong open set condition in the random case

The strong open set condition in the random case

Random Self‐Similar Multifractals

AbstractFor describing the local structure of a random self‐similar measure we use the multi‐fractal decomposition of its support into sets of points of different local dimension. Under the strong open set condition we compute the Hausdorff dimensions of these sets and the generalized dimensions of the random self‐similar measure. Furthermore, the tangential distribution of the random self‐similar measure is investigated.

Separation properties for self-similar sets

Given a self-similar set K in R s {\mathbb {R}^s} we prove that the strong open set condition and the open set condition are both equivalent to H α ( K ) &gt; 0 {H^\alpha }(K) &gt; 0 , where α \alpha is the similarity dimension of K and H α {H^\alpha } denotes the Hausdorff measure of this dimension. As an application we show for the case α = s \alpha = s that K possesses inner points iff it is not a Lebesgue null set.