My papers:

**Geometry of Kalman filters**. . .

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*Abstract*We present a geometric explanation of Kalman filters in terms of a symplectic linear space and a special quadratic form on it. It is an extension of the work of Bougerol \cite{B1} with application of a different metric introduced in \cite{L-W}. The new results are contained in Theorems 1 and 4.

**Gaussian thermostats as geodesic flows of nonsymmetric linear connections**. . .

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*Abstract*We establish that Gaussian thermostats are geodesic flows of special metric connections. We give sufficient conditions for hyperbolicity of geodesic flows of metric connections in terms of their curvature and torsion.

**Design of hyperbolic billiards**. . .

pdf file: 216674 bytes*Abstract*We formulate a general framework for the construction of hyperbolic billiards. Spherical symmetry is exploited for a simple treatment of billiards with spherical caps and soft billiards in higher dimensions. Other examples include the Papenbrock stadium.

**Hyperbolic billiards**. . .

pdf file: 216674 bytes*Abstract*The article published in the Encyclopedia of Mathematical Physics, Elsevier 2006.

**Rigidity of some Weyl manifolds with nonpositive sectional curvature**. . .

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*Abstract*We provide a list of all locally metric Weyl connections with nonpositive sectional curvatures on two types of manifolds, n-dimensional tori $\Bbb T^n$ and $\Bbb M^n =\Bbb S^1\times\Bbb S^{n-1}$ with the standard conformal structures. For $\Bbb M^n$ we prove that it carries no other Weyl connections with nonpositive sectional curvatures, locally metric or not. For $\Bbb T^n$ we prove the same in the more narrow class of integrable connections.

**Rigidity of some Weyl manifolds with nonpositive sectional curvature**. . .

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**Weyl Manifolds and Gaussian Thermostats**. . .

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*Abstract*A relation betwen Weyl connections and Gaussian Thermostats is exposed and exploited.

Published in the Proceedings of the International Congress of Mathematicians, Beijing 2002, Vol III, (2002), 511-523.**Weyl Manifolds and Gaussian Thermostats**. . .

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**Monotonicity, J-algebra of Potapov and Lyapunov exponents**. . .

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*Abstract*We present a new approach and a generalization of the estimates of Lyapunov exponents developed first in \cite{W2} in the symplectic case. The work of Lewowicz \cite{L}, Markarian \cite{M}, and our \cite{W1}, \cite{W2}, \cite{W5}, are combined with the $\Cal J$--algebra of Potapov, \cite{P1},\cite{P2},\cite{P3}. We obtain a general theory which we then specify to the symplectic case. The appendix contains a simple application to the gas of hard spheres.

Published in Smooth Ergodic Theory and Its Applications, Proceedings of Symposia in Pure Mathematics, Vol 69, AMS (2001), 499-521.**W-flows on Weyl manifolds and Gusssian Thermostats**. . .

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*Abstract*We introduce W-flows, by modifying the geodesic flow on a Weyl manifold, and show that they coincide with the isokinetic dynamics. We establish some connections between negative curvature of the Weyl structure and the hyperbolicity of W-flows, generalizing in dimension 2 the classical result of Anosov on Riemannian geodesic flows. In higher dimensions we establish only weaker hyperbolic properties. We extend the theory to billiard W-flows and introduce the Weyl counterparts of Sinai billiards. We obtain that the isokinetic Lorentz gas with the constant external field $E$ and scatterers of radius $r$, studied by Chernov, Eyink, Lebowitz and Sinai in \cite{Ch-E-L-S}, is uniformly hyperbolic, if only $r|E| < 1$, and this condition is sharp.

Published in J. Math. Pures Appl. 79,10 (2000) 953-974.**Magnetic Flows and Gaussian Thermostats**. . .

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*Abstract*We consider a class of flows which include both magnetic flows and Gaussian thermostats of external fields. We give sufficient conditions for such flows on manifolds of negative sectional curvature to be Anosov.

Published in Fundamenta Mathematicae, Vol. 163, (177 -- 191), 2000.**Complete Hyperbolicity in Hamiltonian Systems with Linear Potential and Elastic Collisions**. . .

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*Abstract*We present a class of systems with all Lyapunov exponents different from zero (completely hyperbolic). They are obtained by the restriction of the configuration space of a simple completely integrable system with linear potential and elastic collisions. We show that special geometry of the configuration space is necessary. We survey concrete realizations of this scheme discussed previously in \cite{W6} and elsewhere.

Published in Reports on Mathematical Physics, Vol. 44, (301 -- 312), 1999.**(with Carlangelo Liverani) Conformally Symplectic Dynamics and Symmetry of the Lyapunov Spectrum**. . .

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*Abstract*A generalization of the Hamiltonian formalism is studied and the symmetry of the Lyapunov spectrum established for the resulting systems. The formalism is applied to the Gausssian isokinetic dynamics of interacting particles with hard core collisions and other systems.

Published in Communications in Mathematical Physics, Vol. 194, (47 -- 60), 1998.**Hamiltonian Systems with Linear Potential and Elastic Constraints**. . .

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*Abstract*We consider a class of Hamiltoniansystems with linear potential, elastic constraints and arbitrary number of degrees of freedom. We establish sufficient conditions for complete hyperbolicity of the system.

Published in Fundamenta Mathematicae, Vol. 157, (305 -- 341), 1998.**Hamiltonian Systems with Linear Potential and Elastic Constraints**. . .

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