The shadowing problem is related to the following question: under which condition, for any pseudotrajectory (approximate trajectory) of a vector field there exists a close trajectory? We study $C^1$-interiors of sets of vector fields with various shadowing properties. In the case of discrete dynamical systems generated by diffeomorphisms, such interiors were proved to coincide with the set of structurally stable diffeomorphisms for most general shadowing properties.

We prove that the $C^1$-interior of the set of vector fields with Oriented shadowing property contains not only structurally stable vector fields. Also, we have found additional assumptions under which the $C^1$-interiors of sets of vector fields with Lipschitz, Oriented and Orbit shadowing properties contain only structurally stable vector fields.

Some of these results were obtained together with my advisor S.Yu.Pilyugin.

For an ergodic system, the theorem of Shannon-McMillan-Breiman states that for every finite generating partition the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere. In 1962 Ibragimov showed that the distribution of the measure of cylinder sets is lognormally distributed provided the measure is strong mixing and its conditional entropy function is sufficiently well approximable.
Carleson (1958) and Chung (1960) generalised the theorem of SMB to infinite partitions (provided the entropy is finite). We show that the measures of cylinder sets are lognormally distributed for uniformly strong mixing systems and infinite partitions and show that the rate of convergence is polynomial. Apart from the mixing property we require that a higher than fourth moment of the information function is finite. Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function. We also obtain the law of the iterated logarithm and the weak invariance principle for the information function.