On the structure of quasiminimal sets of foliations on surfaces

Foliations on compact surfaces are considered in this paper. The structure of a quasiminimal set is studied, and criteria for the recurrence of a nonclosed leaf are proved. The concept of an amply situated quasiminimal set is introduced, and the nonexistence of such sets on some orientable and nonorientable surfaces is proved. A sharp estimate of the number of quasiminimal sets of foliations on compact surfaces is given. These results are applied to an estimate of the number of one-dimensional basic sets of $A$-diffeomorphisms of surfaces.