Henk Don

Correlated Fractal Percolation and the Palis Conjecture

Let F1 and F2 be independent copies of one-dimensional correlated fractal percolation, with almost sure Hausdorff dimensions dim H(F1) and dim H(F2). Consider the following question: does dim H(F1)+dim H(F2)>1 imply that their algebraic difference F1−F2 will contain an interval? The well known Palis conjecture states that ‘generically’ this should be true. Recent work by Kuijvenhoven and the first author (Dekking and Kuijvenhoven in J. Eur. Math. Soc., to appear) on random Cantor sets cannot answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condition which permits us to solve the problem, and we prove that the condition of Dekking and Kuijvenhoven (J. Eur. Math. Soc., to appear) implies our condition. Independently of this we give a solution to the critical case, yielding that a strong version of the Palis conjecture holds for fractal percolation and correlated fractal percolation: the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions of the random Cantor sets exceeds one.

Finding DFAs with Maximal Shortest Synchronizing Word Length

It was conjectured by Cerný in 1964 that a synchronizing DFA on n states always has a shortest synchronizing word of length at most \((n-1)^2\), and he gave a sequence of DFAs for which this bound is reached. In 2006 Trahtman conjectured that apart from Cerný’s sequence only 8 DFAs exist attaining the bound. He gave an investigation of all DFAs up to certain size for which the bound is reached, and which do not contain other synchronizing DFAs. Here we extend this analysis in two ways: we drop this latter condition, and we drop limits on alphabet size. For \(n \le 4\) we do the full analysis yielding 19 new DFAs with smallest synchronizing word length \((n-1)^2\), refuting Trahtman’s conjecture. All these new DFAs are extensions of DFAs that were known before. For \(n \ge 5\) we prove that none of the DFAs in Trahtman’s analysis can be extended similarly. In particular, as a main result we prove that the Cerný examples \(C_n\) do not admit non-trivial extensions keeping the same smallest synchronizing word length \((n-1)^2\).

DFAs and PFAs with long shortest synchronizing word length

It was conjectured by Cerný in 1964, that a synchronizing DFA on n states always has a synchronizing word of length at most \((n-1)^2\), and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for \(n \le 4\), and with bounds on the number of symbols for \(n \le 10\). Here we give the full analysis for \(n \le 6\), without bounds on the number of symbols.

Self-averaging sequences which fail to converge

We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that

New methods to bound the critical probability in fractal percolation

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The Černý Conjecture and 1-Contracting Automata

th term is mainly based on terms around a fixed fraction of

Constructing and searching conditioned Galton-Watson trees

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On the distribution of the distances of multiples of an irrational number to the nearest integer

. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.

Metastability of the contact process on Erd\"os-R\'enyi and configuration model graphs

We study the critical probability pcM in two-dimensional M-adic fractal percolation. To find lower bounds, we compare fractal percolation with site percolation. Fundamentally new is the construction of a computable increasing sequence that converges to pcM. We prove that pc2>0.881 and pc3>0.784.