A few references for Spectral Methods in Dynamics

a proof of
the PerronFrobenius
theorem for matrices
by Mike Boyle, U. of
Maryland

proof (with background on BV) of exponential decay of correlations
for piecewiseexpanding maps on the interval: see section 2 in
Gerhard Keller and Carlangelo Liverani: A spectral gap for a
onedimensional lattice of coupled piecewise expanding interval
maps, Lect. Notes Phys. 671 (2005), p. 115151

a clean discussion of exponential decay of correlations for
GibbsMarkov maps (includes fullbranch piecewise expanding interval
maps): see sections 2 (a)(b) in
Ian Melbourne and Matthew
Nicol:
Almost sure invariance principle for nonuniformly hyperbolic
systems, Commun. Math. Phys. 260 (2005) 131146.

Young towers, as introduced by
LaiSang Young (see the
papers on
her publications
page)
LaiSang Young: Statistical properties of dynamical systems with
some hyperbolicity. Ann. of Math. (2) 147 (1998), no. 3,
585650
[introduces Young towers, proves exponential decay of correlations
via spectral methods; includes the quadratic family \(f_a(x):=a x
(1x)\) on [0,1], \(0 < a \le 4\), for which there is a positive
measure set of parameters \(a\) with exponential decay of
correlations; the spectral method implies that such maps have an
acip (absolutely continuous invariant probability), first proved
by Michael Jakobson in 1980]
LaiSang Young: Recurrence times and rates of mixing. Israel
J. Math. 110 (1999), 153188
[proves polynomial decay of correlations, using coupling; includes
the PomeauMannville intermitent maps, extending results of
Carlangelo Liverani, Benoît Saussol and Sandro Vaienti, and
Huyi Hu]
 improved IonescuTulcea & Marinescu
Theorem: see Section II.1 for an introduction and Chapter XIV (Thm.
XIV.3) in
Hubert Hennion and Loic Hervé: Limit Theorems for Markov
Chains and Stochastic Properties of Dynamical Systems by
QuasiCompactness (Lecture Notes in Mathematics 1766, 2001)
 a comprehensive survey, as of 2000, of the topic (beware of typos)
Viviane Baladi: Positive Transfer Operators and Decay of Correlation
(Advanced Series in Nonlinear Dynamics)

similar method/results for subshits of finite type: see the first few
chapters in
William Parry and Mark
Pollicott: Zeta
Functions and the Periodic Orbit Structure of Hyperbolic
Dynamics
The onesided shifts correspond to expanding systems, twosided
shifts are the hyperbolic case.