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Register a new userTransfer operators, atomic decomposition and the Bestiary
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Authors
Daniel Smania
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Arbieto and S. recently used atomic decomposition to study transfer operators. We give a long list of old and new expanding dynamical systems for which those results can be applied, obtaining the quasi-compactness of transfer operator acting on Besov spaces of measure spaces with a good grid.
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We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, the dynamics and the underlying phase space have very low regularity. In particular it is often possible to obtain exponential decay of correlations, the Central Limit Theorem and almost sure invariance principle for fairly general observables, including unbounded ones.
Let $mathcal{R}$ be a strongly compact $C^2$ map defined in an open subset of an infinite-dimensional Banach space such that the image of its derivative $D_F mathcal{R}$ is dense for every $F$. Let $Omega$ be a compact, forward invariant and partially hyperbolic set of $mathcal{R}$ such that $mathcal{R}colon Omegarightarrow Omega$ is onto. The $delta$-shadow $W^s_delta(Omega)$ of $Omega$ is the union of the sets $$W^s_delta(G)= {Fcolon dist(mathcal{R}^iF, mathcal{R}^iG) leq delta, for every igeq 0 },$$ where $G in Omega$. Suppose that $W^s_delta(Omega)$ has transversal empty interior, that is, for every $C^{1+Lip}$ $n$-dimensional manifold $M$ transversal to the distribution of dominated directions of $Omega$ and sufficiently close to $W^s_delta(Omega)$ we have that $Mcap W^s_delta(Omega)$ has empty interior in $M$. Here $n$ is the finite dimension of the strong unstable direction. We show that if $delta$ is small enough then $$cup_{igeq 0}mathcal{R}^{-i}W^s_{delta} (Omega)$$ intercepts a $C^k$-generic finite dimensional curve inside the Banach space in a set of parameters with zero Lebesgue measure, for every $kgeq 0$. This extends to infinite-dimensional dynamical systems previous studies on the Lebesgue measure of stable laminations of invariants sets.
We use the method of atomic decomposition to build new families of function spaces, similar to Besov spaces, in measure spaces with grids, a very mild assumption. Besov spaces with low regularity are considered in measure spaces with good grids, and results on multipliers and left compositions are obtained.
We prove a comprehensive version of the Ruelle-Perron-Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Holder constant of the function generating the operator appears only polynomially, not exponentially as in previous known estimates.
We establish exponential decay in Holder norm of transfer operators applied to smooth observables of uniformly and nonuniformly expanding semiflows with exponential decay of correlations.
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