The Crossed Product by a Partial Endomorphism and the Covariance Algebra

Given a local homeomorphism \sigma:U -> X where U is a clopen subset of an compact and Hausdorff topological space X, we obtain the possible transfer operators L_\rho which may occur for \al:C(X) -> C(U) given by \al(f)=f\sigma. We obtain examples of partial dynamical systems (X_A,\sigma_A) such that the construction of the covariance algebra C^*(X_A,\sigma_A) and the crossed product by partial endomorphism O(X_A,\al,L) associated to this system are not equivalent, in the sense that there does not exists invertible function \rho in C(U) such that O(X_A,\al,L_\rho)=C^*(X_A,\sigma).