The notion of structural stability is one of the oldest concepts in dynamical systems.  Structurally stable diffeomorphisms are conjugate to all nearby diffeomorphisms and are known to be strongly related to dynamics with persistence of the shadowing property.

Motivated by the dynamics of physical phenomena often change (slightly) we study the stability of compositions of hyperbolic dynamical systems and describe conditions under which C1-close sequences of such dynamics are quasi-conjugate or sequentially conjugate.  In particular we obtain a almost sure invariance principle for fastly convergent sequences of Anosov diffeomorphisms or expanding maps, and extend the results on time-dependent Anosov diffeomorphisms . The construction of quasi-conjugacies and sequential conjugacies rely on Lipschitz shadowing properties for non-autonomous dynamical systems and hold for sequences of expanding maps or Anosov diffeomorphisms. Finally, we provide characterizations of the hyperbolicity of invariant foliations for partially hyperbolic diffeomorphisms in terms of a leafwise shadowing property. 

This is a joint work with A. Castro-UFBA and F. Rodrigues-UFRGS.