Electronic Research Announcements

April 2018 , Volume 25

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Zermelo deformation of finsler metrics by killing vector fields
Patrick Foulon and Vladimir S. Matveev
2018, 25: 1-7 doi: 10.3934/era.2018.25.001 +[Abstract](118) +[HTML](61) +[PDF](299.36KB)

We show how geodesics, Jacobi vector fields, and flag curvature of a Finsler metric behave under Zermelo deformation with respect to a Killing vector field. We also show that Zermelo deformation with respect to a Killing vector field of a locally symmetric Finsler metric is also locally symmetric.

Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds
Mohammadreza Molaei
2018, 25: 8-15 doi: 10.3934/era.2018.25.002 +[Abstract](75) +[HTML](32) +[PDF](398.47KB)

We consider a discrete dynamical system on a pseudo-Riemannian manifold and we determine the concept of a hyperbolic set for it. We insert a condition in the definition of a hyperbolic set which implies to the unique decomposition of a part of tangent space (at each point of this set) to two unstable and stable subspaces with exponentially increasing and exponentially decreasing dynamics on them. We prove the continuity of this decomposition via the metric created by a torsion-free pseudo-Riemannian connection. We present a global attractor for a diffeomorphism on an open submanifold of the hyperbolic space \begin{document}$H^2(1)$\end{document} which is not a hyperbolic set for it.

Signatures, sums of hermitian squares and positive cones on algebras with involution
Vincent Astier and Thomas Unger
2018, 25: 16-26 doi: 10.3934/era.2018.25.003 +[Abstract](56) +[HTML](24) +[PDF](379.4KB)

We provide a coherent picture of our efforts thus far in extending real algebra and its links to the theory of quadratic forms over ordered fields in the noncommutative direction, using hermitian forms and "ordered" algebras with involution.

2016  Impact Factor: 0.483




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