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Can every physical system simulate any Turing machine? This is a classical problem that is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore [C. Moore, Nonlinearity 4, 199 (1991)] asked if hydrodynamics is capable of performing computations. More recently, Tao launched a program based on the Turing completeness of the Euler equations to address the blow-up problem in the Navier¿Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem to quantum-field theories. To the best of our knowledge, the existence of undecidable particle paths of three-dimensional fluid flows has remained an elusive open problem since Moore¿s works in the early 1990s. In this article, we construct a Turing complete stationary Euler flow on a Riemannian S3 and speculate on its implications concerning Tao¿s approach to the blow-up problem in the Navier¿Stokes equations., Robert Cardona was supported by the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Program for Units of Excellence in R&D (MDM-2014-0445) via an FPI grant. R.C. and E.M. are partially supported by Grants MTM2015-69135-P/FEDER, the Spanish Ministry of Science and Innovation PID2019-103849GB-I00/AEI/10.13039/501100011033, and Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017SGR932. E.M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. D.P.-S. is supported by MICINN Grant MTM PID2019-106715GB-C21 and MCIU Grant Europa Excelencia EUR2019-103821. F.P. is supported by MICINN/FEDER Grants MTM2016-79400-P and PID2019-108936GB-C21. This work was partially supported by ICMAT–Severo Ochoa Grant CEX2019-000904-S.

For thousands of years, humankind has utilized star tracking to measure both time and geographical location. In the modern technological era, the problem of telling one's orientation and position from images of the stars has newfound importance when related to satellite communication systems. For example, developments in laser-based communication systems promise huge gains in data rates; however, they tend to require a much finer pointing accuracy in order to hit and track their target as compared to radio frequency due to having a more focused emission pattern. As such, satellites with laser-based communications systems require the ability to obtain their attitude with a much higher degree of accuracy than traditional radio-based communication. For example, in [1], the Mars-to-Earth optical communications system studied requires a pointing accuracy on the order of 2-5 microradians, with an estimated update clock of several hundred Hertz. For contrast, the high-gain antenna of the Mars Reconnaissance Orbiter (MRO) had a pointing accuracy requirement of 2.08 milliradians that could update at 10Hz-10kHz [2] (note that MRO did not use star trackers, but rather used the Electra radio; one can study its performance in [3]).

Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The constructed solutions are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three., Comment: 21 pages, 5 figures. Minor corrections, final version to appear at Ergodic Theory and Dynamical Systems

In the first part of this paper we revisit a classical topological theorem by Tischler (1970) and deduce a topological result about compact manifolds admitting a set of independent closed forms proving that the manifold is a fibration over a torus. As an application we reprove the Liouville theorem for integrable systems asserting that the invariant sets or compact connected fibers of a regular integrable system are tori. We give a new proof of this theorem (including the non-commutative version) for symplectic and more generally Poisson manifolds., 8 pages

Tichler proved (Tischler D. 1970 Topology 9, 153-154. (doi:10.1016/0040-9383(70)90037-6)) that a manifold admitting a smooth non-vanishing and closed oneform fibres over a circle. More generally, a manifold admitting k-independent closed one-form fibres over a torus Tk. In this article, we explain a version of this construction for manifolds with boundary using the techniques of b-calculus (Melrose R. 1993 The Atiyah Patodi Singer index theorem. Research Notes in Mathematics. Wellesley, MA: A. K. Peters; Guillemin V, Miranda E, Pires AR. 2014 Adv. Math. (N. Y.) 264, 864-896. (doi:10.1016/j.aim.2014.07.032)). We explore new applications of this idea to fluid dynamics and more concretely in the study of stationary solutions of the Euler equations. In the study of Euler flows on manifolds, two dichotomic situations appear. For the first one, in which the Bernoulli function is not constant, we provide a new proof of Arnold's structure theorem and describe b-symplectic structures on some of the singular sets of the Bernoulli function. When the Bernoulli function is constant, a correspondence between contact structures with singularities (Miranda E, Oms C. 2018 Contact structures with singularities. https://arxiv.org/abs/1806.05638) and what we call b-Beltrami fields is established, thus mimicking the classical correspondence between Beltrami fields and contact structures (see for instance Etnyre J, Ghrist R. 2000 Trans. Am. Math. Soc. 352, 5781-5794. (doi:10.1090/S0002-9947-00-02651-9)). These results provide a new technique to analyse the geometry of steady fluid flows on non-compact manifolds with cylindrical ends., Robert Cardona is supported by FPI-BGSMath doctoral grant. Eva Miranda is supported by the CatalanInstitution for Research and Advanced Studies via an ICREA Academia Prize 2016 and partially supported bythe grants reference number MTM2015-69135-P (MINECO/FEDER) and reference number 2017SGR932 (AGAUR).Daniel Peralta-Salas is supported by the ERC Starting Grant 335079, the MTM grant 2016-76702-P, and partiallysupported by the ICMAT–Severo Ochoa grant SEV-2015-0554. This material is based upon work supported by theNational Science Foundation under Grant No. DMS-1440140 while Eva Miranda was in residence at the MathematicalSciences Research Institute in Berkeley, California, during the Fall 2018 semester.

Moser proved in 1965 in his seminal paper that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group of a manifold coincide. In particular, this yields a classification of compact symplectic surfaces in terms of De Rham cohomology. In this paper we generalize these results for volume forms admitting transversal zeroes. In this case there is also a cohomology capturing the classification: the relative cohomology with respect to the set of critical hypersurface. We compare this classification scheme with the classification of Poisson structures on surfaces which are symplectic away from a hypersurface where they fulfill a transversality assumption ($b$-Poisson structures). We do this using the desingularization technique introduced by Guillemin-Miranda-Weitsman and extend it to $b^m$-Nambu structures., Comment: 10 pages, 2 figures