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Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves. (English) Zbl 1353.37090

In this paper, the author proves a result that is a crucial step towards solving the following problem raised by A. Katok [“Open problems in elliptic dynamics”, http://www.math.psu.edu/katok\_a/problems.html; with D. V. Anosov, Trans. Mosc. Math. Soc. 23, 1–35 (1972; Zbl 0255.58007)]: In low dimensions is every conservative dynamical system with zero topological entropy a limit of integrable systems?
Indeed, the author proves that every \(\mathcal C^\infty\) area-preserving diffeomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic \(\mathcal C^\infty\) diffeomorphisms. The proof uses pseudoholomorphic curve techniques.

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
53D40 Symplectic aspects of Floer homology and cohomology
32Q60 Almost complex manifolds
32Q65 Pseudoholomorphic curves

Citations:

Zbl 0255.58007
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References:

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