Örjan Stenflo

Uniqueness of Invariant Measures for Place-Dependent Random Iterations of Functions

We give a survey of some results within the convergence theory for iterated random functions with an emphasis on the question of uniqueness of invariant probability measures for place-dependent random iterations with finitely many maps. Some problems for future research are pointed out.

Uniqueness in g-measures

Using coupling techniques extending ideas from Harris (1955 Pacific J. Math. 5 707-24), we prove uniqueness in g-measures and give estimates of the rates of convergence for the associated Markov chains, for strictly positive continuous g-functions under a weak regularity condition. Our regularity condition is weaker than the earlier weakest known conditions for uniqueness (Harris T E 1955 Pacific J. Math. 5 707-24; Iosifescu M and Spu

A note on a theorem of Karlin

We give an example of place-dependent random iterations with two affine contractions on the unit interval generating a Markov chain with more than one stationary probability measure. The probability function is continuous and strictly positive. This constitutes a counterexample to a conjecture raised by an incomplete proof by Karlin (Pacific J. Math. 3 (1953) 725-756).

A survey of average contractive iterated function systems

Iterated function systems (IFSs) are useful for creating fractals, interesting probability distributions and enable a unifying framework for analysing stochastic processes with Markovian properties. In this paper, we present a survey of some basic results within the theory of random iterations of functions from an IFS based on average contraction conditions.

Markov chains in random environments and random iterated function systems

We consider random iterated function systems giving rise to Markov chains in random (stationary) environments. Conditions ensuring unique ergodicity and a “pure type” characterization of the limiting “randomly invariant” probability measure are provided. We also give a dimension formula and an algorithm for simulating exact samples from the limiting probability measure.

Ergodic Theorems for Iterated Function Systems Controlled by Regenerative Sequences

Iterated function systems are considered, where the function to iterate in each step is determined by a regenerative sequence. Ergodic theorems of distributional and law of large numbers types are obtained under log-average contractivity conditions.

Dimensions of random affine code tree fractals

We study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random

Iterated function systems with a given continuous stationary distribution

V

A criterion for uniqueness in G-measures and perfect sampling

-variable and homogeneous Markov constructions.

Central limit theorems for contractive Markov chains

For any continuous probability measure μ on ℝ we construct an IFS with probabilities having μ as its unique measure-attractor.