  We study chaotic plane sections of some particular family of triply periodic
surfaces. The question about possible behavior of such sections was posed by S.
P. Novikov. We prove some estimations on the diffusion rate of these sections
using the connection between Novikov's problem and systems of isometries - some
natural generalization of interval exchange transformations. Using
thermodynamical formalism, we construct an invariant measure for systems of
isometries of a special class called the Rauzy gasket, and investigate the main
properties of the Lyapunov spectrum of the corresponding suspension flow.
