Ergodic Theory and Applications in Combinatorial Number Theory

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Proposal: DMS-9971120Principal Investigator: Michael D. BoshernitzanAbstract: The central subject of this proposal is diophantine, combinatorial and spectral properties of iet's (interval exchange transformations) and itm's (interval exchange maps, the nonbijective version of iet's). Boshernitan's approach, unlike the "global" approaches of others, puts special emphasis on the way diophantine and algebraic properties of the parameters involved may affect the dynamical properties of an iet (or an itm). (The diophantine properties considered are often "generic," leading to "global" results.) For some dynamical systems of topological entropy zero, an accurate asymptotic estimate on the corresponding "block growth" is needed that is better than merely subexponential, and in some cases, is even polynomial. These problems are often difficult, more difficult than in the measure-theoretic setting, and many presently available positive results in this direction (of Katok, Yomdin, Gutkin, Haydn, Goetz, and others) are generally believed to be far from best possible. For example, even for itm's, the conjecture on linear block growth (which is supported by computer computations) remains open, and an affirmative resolution of the conjecture would have important consequences. Boshernitzan is also involved in the study of dynamical systems (symbolic flows) of low (in particular, of linear) complexity, as well as various strengthenings of the notion of unique ergodicity tailored to this context, as found in the work of Ferenczi, Cassaigne, Durand, and others. Even though the emphasis of this project is on theoretical questions that arise in the study of dynamical systems of low complexity, there are numerous natural connections with other fields of mathematics (number theory, probability, and combinatorics, to name three), as well as with statistics, computer science (complexity theory, for one) and physics (quantum mechanics). The recent discovery made by Boshernitzan and his collaborator Kornfeld of "exotic" interval translation maps may prove to have applications in laser technology.