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Conditions for Equality between Lyapunov and Morse Decompositions
Let be a continuous principal bundle whose group is
reductive. A flow of automorphisms of endowed with an ergodic
probability measure on the compact base space induces two decompositions of
the flag bundles associated to . A continuous one given by the finest Morse
decomposition and a measurable one furnished by the Multiplicative Ergodic
Theorem. The second is contained in the first. In this paper we find necessary
and sufficient conditions so that they coincide. The equality between the two
decompositions implies continuity of the Lyapunov spectra under pertubations
leaving unchanged the flow on the base space
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
A remarkable number of different numerical algorithms can be understood and
analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant
curvature and their tangents. This theory can be used to unify a range of
different topics, such as polar-type matrix decompositions, splitting methods
for computation of the matrix exponential, composition of selfadjoint numerical
integrators and dynamical systems with symmetries and reversing symmetries. The
thread of this paper is the following: involutive automorphisms on groups
induce a factorization at a group level, and a splitting at the algebra level.
In this paper we will give an introduction to the mathematical theory behind
these constructions, and review recent results. Furthermore, we present a new
Yoshida-like technique, for self-adjoint numerical schemes, that allows to
increase the order of preservation of symmetries by two units. Since all the
time-steps are positive, the technique is particularly suited to stiff
problems, where a negative time-step can cause instabilities
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