The regularity of central leaves of partially hyperbolic sets and its applications

We consider partially hyperbolic maps which are close to the direct product of a hyperbolic map and an identity map and prove that their central leaves depend Hölder continuously on the base point in the $C^r$-metric. We use this result to construct an open set of diffeomorphisms with rather unusual properties (they have transitive sets with periodic points of different indices and orbits with zero Lyapunov exponent). This paper concludes a series of joint papers with Yu. S. Ilyashenko.