Description of Generalized Continued Fractions by Finite Automata

A generalized continued fraction algorithm associates with every real number x a sequence of integers; x is rational iff the sequence is finite. For a fixed algorithm, call a sequence of integers valid if it is the result of that algorithm on some input x0. We show that, if the algorithm is sufficiently well-behaved, then the set of all valid sequences is accepted by a finite automaton. I. Introduction. It is well known that every real number x has a unique expansion as a simple continued fraction in the form

Continued fractions and transcendental numbers

It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of the present work is to present new families of transcendental continued fractions with bounded partial quotients. Our results are derived thanks to new combinatorial transcendence criteria recently obtained by Adamczewski and Bugeaud

Decidability Problems for Self-induced Systems Generated by a Substitution

International audienceIn this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems

Boolean Delay Equations: A simple way of looking at complex systems

Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the classification of ordinary or partial differential equations. Solutions to certain conservative BDEs exhibit growth of complexity in time. They represent therewith metaphors for biological evolution or human history. Dissipative BDEs are structurally stable and exhibit multiple equilibria and limit cycles, as well as more complex, fractal solution sets, such as Devil's staircases and ``fractal sunbursts``. All known solutions of dissipative BDEs have stationary variance. BDE systems of this type, both free and forced, have been used as highly idealized models of climate change on interannual, interdecadal and paleoclimatic time scales. BDEs are also being used as flexible, highly efficient models of colliding cascades in earthquake modeling and prediction, as well as in genetics. In this paper we review the theory of systems of BDEs and illustrate their applications to climatic and solid earth problems. The former have used small systems of BDEs, while the latter have used large networks of BDEs. We moreover introduce BDEs with an infinite number of variables distributed in space (``partial BDEs``) and discuss connections with other types of dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions.Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular the discussion on partial BDEs is updated and enlarge

Distributing Labels on Infinite Trees

Sturmian words are infinite binary words with many equivalent definitions: They have a minimal factor complexity among all aperiodic sequences; they are balanced sequences (the labels 0 and 1 are as evenly distributed as possible) and they can be constructed using a mechanical definition. All this properties make them good candidates for being extremal points in scheduling problems over two processors. In this paper, we consider the problem of generalizing Sturmian words to trees. The problem is to evenly distribute labels 0 and 1 over infinite trees. We show that (strongly) balanced trees exist and can also be constructed using a mechanical process as long as the tree is irrational. Such trees also have a minimal factor complexity. Therefore they bring the hope that extremal scheduling properties of Sturmian words can be extended to such trees, as least partially. Such possible extensions are illustrated by one such example.Comment: 30 pages, use pgf/tik

Processes with Long Memory: Regenerative Construction and Perfect Simulation

We present a perfect simulation algorithm for stationary processes indexed by Z, with summable memory decay. Depending on the decay, we construct the process on finite or semi-infinite intervals, explicitly from an i.i.d. uniform sequence. Even though the process has infinite memory, its value at time 0 depends only on a finite, but random, number of these uniform variables. The algorithm is based on a recent regenerative construction of these measures by Ferrari, Maass, Mart{\'\i}nez and Ney. As applications, we discuss the perfect simulation of binary autoregressions and Markov chains on the unit interval.Comment: 27 pages, one figure. Version accepted by Annals of Applied Probability. Small changes with respect to version

Coinductive counting with weighted automata

A general methodology is developed to compute the solution of a wide variety of basic counting problems in a uniform way: (1) the objects to be counted are enumerated by means of an infinite weighted automaton; (2) the automaton is reduced by means of the quantitative notion of stream bisimulation; (3) the reduced automaton is used to compute