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]

the
original IonescuTulcea & Marinescu Theorem: Théorie
ergodique pour des classes d’opérations non complètement continues,
C. T. Ionescu Tulcea and G. Marinescu,
Annals of Mathematics (2),
52 (July 1950), 140147.
Here is the Mathscinet review.
 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.

"Further reading", which explains how to deal with
hyperbolic systems without reducing first to the expanding direction.
To obtain a spectral gap in this setting, different Banach spaces are
needed. The paper starts with a very illuminating example of a
contracting system.
Mark Demers:
A gentle introduction to anisotropic Banach spaces,,
Chaos, Solitons and Fractals 116 (2018), 2942.
preprint
at http://faculty.fairfield.edu/mdemers/research/2018.09.12.normsurvey.proofed.pdf