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Đề tài " A C2-smooth counterexample to the Hamiltonian Seifert conjecture in R4 "

We construct a proper C 2 -smooth function on R4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C 2 -smooth counterexample to the Hamiltonian Seifert conjecture in dimension four. 1. Introduction The “Hamiltonian Seifert conjecture” is the question whether or not there exists a proper function on R2n whose Hamiltonian flow has no periodic orbits on at least one regular level set. | Annals of Mathematics A C2-smooth counterexample to the Hamiltonian Seifert conjecture in R4 By Viktor L. Ginzburg and Basnak Z. G urel Annals of Mathematics 158 2003 953 976 A C2-smooth counterexample to the Hamiltonian Seifert conjecture in R4 By Viktor L. Ginzburg and Basak Z. Gurel Abstract We construct a proper C2-smooth function on R4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four. 1. Introduction The Hamiltonian Seifert conjecture is the question whether or not there exists a proper function on R2ra whose Hamiltonian flow has no periodic orbits on at least one regular level set. We construct a C2-smooth function on R4 with such a level set. Following the tradition of Gi4 He1 He2 Ke KuG KuGK KuK1 KuK2 Sc we can call this result a C2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four. We emphasize that in this example the Hamiltonian vector field is C 1-smooth while the function is C2 . In dimensions greater than six C -smooth counterexamples to the Hamiltonian Seifert conjecture were constructed by one of the authors Gi1 and simultaneously by M. Herman He1 He2 . In dimension six a C21 -smooth counterexample was found by M. Herman He1 He2 . This smoothness constraint was later relaxed to C x in Gi2 . A very simple and elegant construction of a new C x-smooth counterexample in dimensions greater than four was recently discovered by E. Kerman Ke . The flow in Kerman s example has dynamics different from the ones in Gi1 Gi2 He1 He2 . We refer the reader to Gi3 Gi4 for a detailed discussion of the Hamiltonian Seifert conjecture. The reader interested in the results concerning the original Seifert conjecture settled by K. Kuperberg KuGK KuK1 should consult KuK2 KuK3 . Here we only mention that a C 1-smooth counterexample to the Seifert conjecture on S3 was constructed by P. Schweitzer Sc