We consider iterates of absolutely continuous measures concentrated in a neighbourhood of a partially hyperbolic attractor. It is shown that limit points can be measures which have conditional measures of a special form for any partition into subsets of unstable manifolds.
We introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.
We study the stability of motion in the form of travelling waves in lattice models of unbounded multi-dimensional and multi-component media with a nonlinear prime term and small coupling depending on a finite number of space coordinates.Under certain conditions on the nonlinear term we show that the set of travelling waves running with the same sufficiently large velocity forms a finite-dimensional submanifold in infinite-dimensional phase space endowed with a special metric with weights. It is 'almost' stable and contains a finitedimensional strongly hyperbolit subset invariant under both evolution operator and space translations.
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