This book presents the reader with a streamlined exposition of the notions and results leading
to the construction of normal forms and ultimately to the construction of smooth conjugacies
for the perturbations of tempered exponential dichotomies. These are exponential dichotomies
for which the exponential growth rates of the underlying linear dynamics never vanish. In other
words its Lyapunov exponents are all nonzero. The authors consider mostly difference equations
although they also briefly consider the case of differential equations. The content is
self-contained and all proofs have been simplified or even rewritten on purpose for the book so
that all is as streamlined as possible. Moreover all chapters are supplemented by detailed
notes discussing the origins of the notions and results as well as their proofs together with
the discussion of the proper context also with references to precursor results and further
developments. A useful chapter dependence chart is included in the Preface. The book is aimed
at researchers and graduate students who wish to have a sufficiently broad view of the area
without the discussion of accessory material. It can also be used as a basis for graduate
courses on spectra normal forms and smooth conjugacies. The main components of the exposition
are tempered spectra normal forms and smooth conjugacies. The first two lie at the core of
the theory and have an importance that undoubtedly surpasses the construction of conjugacies.
Indeed the theory is very rich and developed in various directions that are also of interest
by themselves. This includes the study of dynamics with discrete and continuous time of
dynamics in finite and infinite-dimensional spaces as well as of dynamics depending on a
parameter. This led the authors to make an exposition not only of tempered spectra and
subsequently of normal forms but also briefly of some important developments in those other
directions. Afterwards the discussion continues with the construction of stable and unstable
invariant manifolds and consequently of smooth conjugacies while using most of the former
material. The notion of tempered spectrum is naturally adapted to the study of nonautonomous
dynamics. The reason for this is that any autonomous linear dynamics with a tempered
exponential dichotomy has automatically a uniform exponential dichotomy. Most notably the
spectra defined in terms of tempered exponential dichotomies and uniform exponential
dichotomies are distinct in general. More precisely the tempered spectrum may be smaller
which causes that it may lead to less resonances and thus to simpler normal forms. Another
important aspect is the need for Lyapunov norms in the study of exponentially decaying
perturbations and in the study of parameter-dependent dynamics. Other characteristics are the
need for a spectral gap to obtain the regularity of the normal forms on a parameter and the
need for a careful control ofthe small exponential terms in the construction of invariant
manifolds and of smooth conjugacies.
