The problem of gray boxes (Q918014)

The present note is a continuation of the investigation of the author [Funct. Anal. Appl. 20, 51-52 (1986); translation from Funkts. Anal. Prilozh. 20, No. 1, 62-63 (1986; Zbl 0625.58030) and Methods Qualitative Theory Diff. Equations, Gorki State Univ., 3-18 (1985)] on the problem of gray boxes of Cherry flows on the torus \(T^ 2\), posed in 1937 by \textit{T. M. Cherry} [Proc. Lond. Math. Soc., II. Ser. 44, 175-215 (1938; Zbl 0019.11503)] in connection with the solution of the Poincaré conjecture for quasiminimal sets, i.e., for the sets that are closures of nonclosed Poisson-stable nonrecurrent trajectories. \textit{T. M. Cherry} [loc. cit.] has constructed an analytic flow with one black box on \(T^ 2\) that has a nowhere dense quasiminimal set and, by the same token, has shown that if the Poincaré conjecture is interpreted in a wider sense than that of Poincaré, then it has an affirmative solution although the problem of structure of the quasiminimal set \(\Omega\) itself, namely the so-called problem of gray boxes, has remained unsolved. But if we remain within the framework of the Poincaré conjecture (there exists an analytic flow \(\Phi^ t\) on \(T^ 2\) without equilibrium states that has a nowhere dense minimal set, different from a closed trajectory), then \textit{A. Denjoy} [J. Math. Pure Appl., IX. Sér. 11, 333-375 (1932; Zbl 0006.30501)] obtained a negative solution not only for analytic, but even for \(C^ 2\)-flows (the Poincaré conjecture is valid for \(C^ 1\)-flows).