Local Limit Theorems for Partial Sums of Stationary Sequences Generated by Gibbs–markov Maps

Abstract

We introduce Gibbs–Markov maps T as maps with a (possibly countable) Markov partition and a certain type of bounded distortion property, and investigate its Frobenius–Perron operator P acting on (locally) Lipschitz continuous functions ϕ. If such a function ϕ belongs to the domain of attraction of a stable law of order in (0,2), we derive the expansion of the eigenvalue function t↦λ(t) of the characteristic function operators Ptf=Pf exp [i< t,ϕ> (perturbations of P) around 0. From this representation local and distributional limit theorems for partial sums ϕ+…+ϕ◦ Tn are easily obtained, provided ϕ is aperiodic. Applications to recurrence properties of group extensions are also given.

S-unimodal Misiurewicz maps with flat critical points

“…A real-valued random variable R (respectively its distribution) is called stable (cf. [AD1] or [IL]) if for all a, b > 0 there are c > 0 and d ∈ R such that aR + bR * d = cR + d, where R * is an independent copy of R and R d = S means equality of distributions. In this case a α + b α = c α for some α ∈ (0, 2], called Remark 7 (Dynkin-Lamperti arcsin law).…”

“…Applying the Koebe principle relative to I M to small intervals (x, y) in Y shows that the induced system (Y, T Y , ξ Y ) in (6) in fact has the Gibbs property in the sense of [A0], [AD1].…”

Multidimensional local central limit theorem of some non-uniformly hyperbolic systems

“…To this end, we need the following result which is Proposition 3.7 in [20]. The proof of this lemma can be found in [20].…”

Smooth Livšic regularity for piecewise expanding maps

“…For mixing Lasota-Yorke maps and Sinai dispersing billiards they show that such a χ is Lipschitz on an open set. There is an error in [15,Theorem 1] in the setting of C 2 Markov maps -they only prove measurable solutions χ to Equation (1) are Lipschitz on each element T α, α ∈ P, where P is the defining partition for the Markov map, and not that the solutions are Lipschitz on X, as Theorem 1 erroneously states. The error arose in the following way: if χ is Lipschitz on α ∈ P it is possible to extend χ as a Lipschitz function to T α by defining χ(T x) = φ(x) + χ(x), however extending χ as a Lipschitz function from α to T 2 α via the relation χ(T 2 x) = φ(T x) + χ(T x) may not be possible, as φ • T may have discontinuities on T α.…”

“…In related work, Aaronson and Denker [1,Corollary 2.3] have shown that if (T, X, µ, P) is a mixing Gibbs-Markov map with countable Markov partition P preserving a probability measure µ and φ : X → R d is Lipschitz (with respect to a metric ρ on X derived from the symbolic dynamics) then any measurable solution χ : X → R d to φ = χ • T − χ has a versionχ which is Lipschitz continuous, i.e. there exists C > 0 such that d(χ(x),χ(y)) ≤ Cρ(x, y) for all x, y ∈ T (α) and each α ∈ P.…”