Summary of Publications and Preprints

Call diffeomorphisms satisfying (2) Artin-Mazur (A-M) diffeomorphisms. In [I.1] we give an elementary proof of an extension of Artin–Mazur’s result, namely, we prove that A-M diffeomorphisms with only hyperbolic periodic points are dense in Diffr(M). In [I.2] we construct Baire generic diffeomorphisms inside of an open set (a Newhouse domain) in Diffr(M) of with superexponential (even an arbitrary fast) growth of number of periodic points. This theorem is based on a theorem of GonchenkoShilnikov-Turaev, which is also rigorously proved in [I.2]. In [I.3] we outline ideas from the proofs in [I.1], [I.2], [I.5 -I.6], and prove some new results. We call a set of diffeomorphisms prevalent if in a sense it has “probability one”. In [I.5 -I.6] we prove that for a prevalent diffeomorphism (or with “probability 1”) for any positive 2 > 0 there is C = C(2, f) such that Pn(f) < exp(Cn) for any n ∈ Z+. (3)