Invariant measures and their properties. A functional analytic point of view

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ALMOST SURE INVARIANCE PRINCIPLE FOR DYNAMICAL SYSTEMS BY SPECTRAL METHODS (2009, background): “ we have kLn t kC→C ≤CMn; 2. for all n∈ N, all |t| ≤ ε 0 and all u∈ B, we have kLn t ukB ≤ CκnkukB + CMnkukC; 3. the quantity kLt −L 0kB→C tends to 0 when t→0. Then Keller and Liverani [14], Liverani [15] show that, for small enough t, the operator Lt acting on B has a simple eigenvalue λ(t) close t”
Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations (2006, background): “A bound like (2) is usually referred to as the Lasota–Yorke inequality [30, 31] or the Ionescu–Tulcea–Marinescu inequality [26] and is used to show the existence of absolutely continuous invariant measures when Pt is the transfer operator acting on densities.”
Lorenz-like flows: exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence (2009, background): “The above lemmata give the following result, in the spirit of the Lasota–Yorke inequality (see in the following proof, equation (A5) and compare with, for example, [24, 25]) giving an upper bound on the variation of iterates F(μ0).”