Generalized counterexamples to the Seifert conjecture

Hamiltonian Dynamical Systems Without Periodic Orbits

The present paper is a review of counterexamples to the ``Hamiltonian Seifert conjecture'' or, more generally, of examples of Hamiltonian systems having no periodic orbits on a compact energy level. We begin with the discussion of the ``classical'' and volume--preserving Seifert conjectures. Then a construction of counterexamples to the Hamiltonian Seifert conjecture in dimensions greater than or equal to six is outlined, mainly following the method introduced by the author of the paper. The Hamiltonian version of Seifert's theorem is also stated. A list of all known at this moment examples an…

Seifert conjecture in the even convex case

In this paper, we prove that there exist at least $n$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface $\Sg$ in $\R^{2n}$ satisfying the reversible condition $N\Sg=\Sg$ with $N=\diag (-I_n,I_n)$. As a consequence, we show that if the Hamiltonian function is convex and even, then Seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integer $n$.

Real analytic counterexample to the freeness conjecture

We provide a counterexample to P.~Olver's freeness conjecture for $C^ω$ transformations.

A Computational Method for Large Scale Searching of Counterexamples to the Strong Coincidence Conjecture

In this short note we report on results on a computational search for a counterexample to the strong coincidence conjecture. In particular, we discuss the method used so that further searches can be conducted.

The Hamiltonian Seifert Conjecture: Examples and Open Problems

Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is the existence problem for regular compact energy levels without periodic orbits. Very little is known about how large the set of regular energy values without periodic orbits can be. For instance, in all known examples of Hamiltonian flows on linear spaces such energy values form a discrete set, whereas ``almost existence theorems'' only require this set to…

Extended Fuller index, sky catastrophes and the Seifert conjecture

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields $X$ on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at $X$ has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on $S ^{2k+1} $. A sky catastrophe, is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller's foundational papers. We also put strong constraints on t…

A counterexample to the "composition conjecture"

In this note we construct a class of counterexamples to the "composition conjecture" concerning an infinitesimal version of the center problem for the polynomial Abel equation in the complex domain.

双曲度量附近普通Fried猜想的反例

Counterexamples to the ordinary Fried conjecture near hyperbolic metrics

该研究借助LLM辅助构造三维双曲度量附近普通Fried猜想的反例，证明非循环酉线特征趋于平凡特征时，Ruelle ζ函数零点模与解析挠率平方的比值趋于0。