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Dynamics retrospective: Great problems, attempts that failed. (English) Zbl 0745.58018
The author himself considers this note as “reflecting a voice out of the past”, “the past” that is the period of his activity in dynamics, the most part of which ended before 1970. This voice selects the following ten problems of dynamical systems on a compact state space \(M\).
I. Is the dynamics of the Lorenz differential equations in \(\mathbb{R}^ 3\) described by “geometric Lorenz attraction” of Williams, Guckenheimer and Yorke?
II. Can the Navier-Stokes equations on the two-dimensional torus be dynamically non-trivial?
III. Is Anosov diffeomorphism \(T\) topologically the same as the Lie group model of John Franks?
IV. Is axiom A a generic property for one-dimensional dynamical systems? (Two cases: a) real; b) complex).
V. It is generic property of \(T\in \hbox{Diff}(M)\) (\(C^ r\), \(r>1\)) that the periodic points are dense in the non-wondering points?
VI. It is generic property that the centralizer \(Z(T)\) of \(T\in \hbox{Diff}(M)\) consists of only the iterates of \(T\)?
VII. Given two real polynomials in two variables \(P\) and \(Q\) and the differential equations on \(\mathbb{R}^ 2\), \[ dx/dt=P(x,y), dy/dt=Q(x,y), \] is there a bound \(K\) of the number of limit cycles of the form \(K\leq n^ q\), where \(n\) is the maximum of the degree of \(P\) and \(Q\)?
VIII. Given any set of masses, \(m_ 1,\dots,m_ n>0\) in the \(n\)-body problem of celestial mechanics, is the number of relative equilibria finite?
IX. Extend the mathematical model of general equilibrium theory to include price adjustments.
X. Extend the dynamics of quantum mechanics in a way which contains the successful features of the existing theory, and permits transitions between equilibrium states.
Also a short discussion of each is given with some references.

37Cxx Smooth dynamical systems: general theory
35Q30 Navier-Stokes equations
Full Text: DOI
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