F. M. Dekking

The spectrum of dynamical systems arising from substitutions of constant length

SummaryMinimal flows and dynamical systems arising from substitutions are considered. In the case of substitutions of constant length the trace relation of the flow is calculated and is used to determine the spectrum of the dynamical system. Several methods are indicated to obtain new substitutions from given ones, leading among other things to a description of the behaviour of powers of the shift homeomorphism on the system arising from any substitution.

Strongly non-repetitive sequences and progression-free sets

Abstract By a simple method we show the existence of (1) a sequence on two symbols in which no four blocks occur consecutively that are permutations of each other, and (2) a sequence on three symbols in which no three blocks occur consecutively that are permutations of each other. The problem of the existence of a sequence on four symbols in which no two blocks occur consecutively that are permutations of each other remains open.

On the structure of Mandelbrot's percolation process and other random cantor sets

We consider generalizations of Mandelbrots percolation process. For the process which we call the random Sierpinski carpet, we show that it passes through several different phases as its parameter increases from zero to one. The final section treats the percolation phase.

Superbranching processes and projections of random Cantor sets

AbstractWe study sequences (X0, X1, ...) of random variables, taking values in the positive integers, which grow faster than branching processes in the sense that

Limit distributions for minimal displacement of branching random walks

X_{m + n} \geqq \sum\limits_{i = 1}^{X_m } {X_n (m,i)}

On repetitions of blocks in binary sequences

, for m, n≧0, where the Xn(m, i) are distributed as Xn and have certain properties of independence. We prove that, under appropriate conditions, Xn1/n→λ almost surely and in L1, where λ=sup E(Xn)1/n. Our principal application of this result is to study the Lebesgue measure and (Hausdorff) dimension of certain projections of sets in a class of random Cantor sets, being those obtained by repeated random subdivisions of the M-adic subcubes of [0, 1]d. We establish a necessary and sufficient condition for the Lebesgue measure of a projection of such a random set to be non-zero, and determine the box dimension of this projection.

Simulation of Rough, Elastic Contacts

SummaryMeasure-theoretical and topological mixing properties of dynamical systems arising from substitutions are considered. It is shown that such systems are neither measure-theoretically strongly mixing, nor topologically strongly mixing of all orders. In the case of the substitution