Dimension and invertibility of hyperbolic endomorphisms with singularities
- Schmeling, Jörg
- Troubetzkoy, Serge
2010 Mathematics Subject Classification
- cocycles, transfer map, Anosov system, hyperbolic attractors, existence of SBR measures, Young dimension formula
We introduce a class of endomorphisms which are piecewise smooth and have hyperbolic attractors. This class generalizes the class of piecewise smooth diffeomorphisms with hyperbolic attractors studied by Pesin , Sataev , and others . Examples in our class are the fat Belykh map, projections of Solenoids onto cross-sections, and crossed horseshoes. We first develop the stable manifold theory, the existence of SBR measures and the ergodic theory of our class of maps. This theory mostly parallels the invertible case so we only sketch some of the important arguments. We generally follow the outline of , details can be found there. Our main results, theorem 5.2 hold in the two dimensional case: if the product of the Lyapunov exponents is less than one the mapping being invertible µSBR-a.e. on the attractor is equivalent to the Young formula holding. If the mapping is not invertible a.e. we can calculate the defect in the dimension formula 5.4. If the product of the Lyapunov exponents is greater than one then the attractor is two dimensional and the mapping restricted to the attractor is not invertible on a set of positive measure. Finally, we also give an easily checkable sufficient condition for a map to belong to the general class of maps we consider. This condition is easy to check for the systems we were motivated. In particular, in  this theorem is applied to fat Belykh maps where the entire picture of everywhere invertibility, invertibility on the attractor, almost everywhere invertibility on the attractor and noninvertibility almost surely is understood. Kaplan and Yorke have conjectured that for a broad class of systems the dimension of the attractor equals the Lyapunov dimension for most maps from the class. The results of  imply that for the Belykh family the Kaplan-Yorke conjecture holds for almost all parameter values.
- Ergodic Theory Dynam. Systems 18 (1998) no. 5, pp. 1257--1282.