In this book the theory of hyperbolic sets is developed, both for
diffeomorphisms and flows, with an emphasis on shadowing. We show
that hyperbolic sets are expansive and have the shadowing property.
Then we use shadowing to prove that hyperbolic sets are robust
under perturbation, that they have an asymptotic phase property and
also that the dynamics near a transversal homoclinic orbit is
chaotic. It turns out that chaotic dynamical systems arising in
practice are not quite hyperbolic. However, they possess enough
hyperbolicity to enable us to use shadowing ideas to give
computer-assisted proofs that computed orbits of such systems can
be shadowed by true orbits for long periods of time, that they
possess periodic orbits of long periods and that it is really true
that they are chaotic. Audience: This book is intended primarily
for research workers in dynamical systems but could also be used in
an advanced graduate course taken by students familiar with
calculus in Banach spaces and with the basic existence theory for
ordinary differential equations.
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