  In this paper we further explore the L-shadowing property defined in [17] for
dynamical systems on compact spaces. We prove that structurally stable
diffeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional
sphere satisfy this property. Homeomorphisms satisfying the L-shadowing
property have a spectral decomposition where the basic sets are either
expansive or contain arbitrarily small topological semi-horseshoes (periodic
sets where the restriction is semiconjugate to a shift). To this end, we
characterize the L-shadowing property using local stable and unstable sets and
the classical shadowing property. We exhibit homeomorphisms with the
L-shadowing property and arbitrarily small topological semi-horseshoes without
periodic points. At the end, we show that positive finite-expansivity jointly
with the shadowing property imply that the space is finite.
