# A few references for Spectral Methods in Dynamics

• a proof of the Perron-Frobenius theorem for matrices by Mike Boyle, U. of Maryland
• proof (with background on BV) of exponential decay of correlations for piecewise-expanding maps on the interval: see section 2 in
Gerhard Keller and Carlangelo Liverani: A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps, Lect. Notes Phys. 671 (2005), p. 115-151
• a clean discussion of exponential decay of correlations for Gibbs-Markov maps (includes full-branch 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) 131-146.
• Young towers, as introduced by Lai-Sang Young (see the papers on her publications page)
Lai-Sang Young: Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 (1998), no. 3, 585-650
[introduces Young towers, proves exponential decay of correlations via spectral methods; includes the quadratic family $$f_a(x):=a x (1-x)$$ 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]

Lai-Sang Young: Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153-188
[proves polynomial decay of correlations, using coupling; includes the Pomeau-Mannville intermitent maps, extending results of Carlangelo Liverani, Benoît Saussol and Sandro Vaienti, and Huyi Hu]
• the original Ionescu-Tulcea & 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), 140-147.

Here is the Mathscinet review.
• improved Ionescu-Tulcea & 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 Quasi-Compactness (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 one-sided shifts correspond to expanding systems, two-sided 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), 29-42.

preprint at http://faculty.fairfield.edu/mdemers/research/2018.09.12.normsurvey.proofed.pdf