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We investigate a class of homeomorphisms of a cylinder, with all trajectories convergent to the cylinder base and one fixed point in the base. Let A be a nonempty finite or countable family of sets, each of which can be a priori an $\omega$-limit set. Then there is a homeomorphism from our class, for which A is the family of all $\omega$-limit sets.
keywords: $\omega$-limit set.
In [PS] it is conjectured that among the volume preserving $C^2$ diffeomorphisms of a closed manifold which have some hyperbolicity, the ergodic ones contain an open and dense set. In this paper we prove an analogous statement for skew products of Anosov diffeomorphisms of tori and circle rotations. Thus this paper may be seen as an example of the phenomenon conjectured in [PS]. The corresponding theorem for skew products of Anosov diffeomorphisms and translations of arbitrary compact groups is an interesting open problem.
We investigate a family of maps that arises from a model in economics and game theory. It has some features similar to renormalization and some similar to intermittency. In a one-parameter family of maps in dimension 2, when the parameter goes to 0, the maps converge to the identity. Nevertheless, after a linear rescaling of both space and time, we get maps with attracting invariant closed curves. As the parameter goes to 0, those curves converge in a strong sense to a certain circle. We call those phenomena microdynamics. The model can be also understood as a family of discrete time approximations to a Brown-von Neumann differential equation.
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