The wavelet Galerkin operator appears in several different contexts such as wavelets (see for example [Law91b], [Dau95], [CoDa96], [LLS96], [CoRa90]), ergodic theory and g-measures ([Kea82]) or quantum statistical mechanics ([Rue68]). For some of the applications of the Ruelle operator we refer the reader to the book by V. Baladi [Bal00]. It also bears many different names in the literature: th...

Let G be a group and $$\ell $$ commutative unital $$*$$ -ring with an element $$\lambda \in \ell such that + \lambda ^*= 1$$ . We introduce variants of hermitian bivariant K-theory for -algebras equipped G-action or G-grading. For any graph E finitely many vertices weight function $$\omega :E^1 \rightarrow G$$ , distinguished triangle $$L(E)=L_\ell (E)$$ in the G-graded category $$kk^h_{G_{{\te...

We show that there are no partially hyperbolic horseshoes with positive Lebesgue measure for diffeomorphisms whose class of differentiability is higher than 1. This generalizes a classical result by Bowen for uniformly hyperbolic horseshoes.