We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also ...

We describe a construction of a moduli space of solenoid functions for the C -conjugacy classes of hyperbolic dynamical systems f on surfaces with hyperbolic basic sets f . We explain that if the holonomies are sufficiently smooth then the diffeomorphism f is rigid in the sense that it is C 1C conjugate to a hyperbolic affine model. We present a moduli space of measure solenoid functions for al...

A companion for the Ostrowski and the generalised trapezoid inequalites for various classes of functions, including functions of bounded variation, Lipschitzian, convex and absolutely continuous functions is established. Applications for weighted means are also given.

In this note, the notion of absolute continuity of functions of two variables is discussed. We recall that the set of functions of two variables absolutely continuous in the sense of Carathéodory coincides with the class of functions admitting a certain integral representation. We show that absolutely continuous functions in the sense of Carathéodory can be equivalently characterized in terms o...