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Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined— " chaos. " Mathematicians quickly pointed out that the purported revolution was relying on the… (More)
Brillouin zones were introduced by Brillouin [Br] in the thirties to describe quantum mechanical properties of crystals, that is, in a lattice in R n. They play an important role in solid-state physics. It was shown by Bieberbach [Bi] that Brillouin zones tile the underlying space and that each zone has the same area. We generalize the notion of Brillouin… (More)
In this note we consider the well known Hannay-Ozorio de Almeida sum formula from a mathematically rigorous viewpoint. In particular, we discuss situations where we can obtain the Sinai-Ruelle-Bowen measure as a limit taken over periodic orbits with periods in an interval which shrinks as it moves to infinity.
In Kupka et al. 2006 appears the Focal Stability Conjecture: the focal decomposition of the generic Riemann structure on a manifold M is stable under perturbations of the Riemann structure. In this paper, we prove the conjecture when M has dimension two, and there are no conjugate points.