4,137 research outputs found
Mildly dissipative diffeomorphisms of the disk with zero entropy
We discuss the dynamics of smooth diffeomorphisms of the disc with vanishing
topological entropy which satisfy the mild dissipation property introduced in
[CP]. In particular it contains the H\'enon maps with Jacobian up to 1/4. We
prove that these systems are either (generalized) Morse Smale or infinitely
renormalizable. In particular we prove for this class of diffeomorphisms a
conjecture of Tresser: any diffeomorphism in the interface between the sets of
systems with zero and positive entropy admits doubling cascades. This
generalizes for these surface dynamics a well known consequence of
Sharkovskii's theorem for interval maps
A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations on and those
of the parabolic equations on a bounded
domain . We give details on the similarities of these dynamics in the
cases , and and in the corresponding cases ,
and dim() respectively. In addition to
the beauty of such a correspondence, this could serve as a guideline for future
research on the dynamics of parabolic equations
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