On mixing and the local central limit theorem for hyperbolic flows

Abstract

We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal processes, Axiom A flows, as well as the systems admitting Young tower, such as Sinai billiard with finite horizon, suspensions over Pomeau-Manneville maps, and geometric Lorenz attractors. t 0 ϕ(Φ s (x))ds with respect to an infinite invariant measure µ×Leb. Also, in the discrete case, MLCLT has an interpretation as mixing of a certain Z extension of Φ (see [AN17]).The result of our analysis is that (1.1) may in general fail for some arithmetic reasons (see Section 6.1 for an explicit example). However all limit points of the LHS of (1.1) are of the form given by the RHS of (1.1) with, possibly, different measures u. We also provide sufficient conditions for (1.1) as well as for MLCLT to hold.

Infinite measure renewal theorem and related results

“…In other words, if the appropriate local limit theorem and large deviation bounds hold for the base map, then the special flow is mixing in both aperiodic and periodic irrational cases but not in the rational case. A similar result holds in the finite measure case (see [, Section 2]).…”

Expansions in the Local and the Central Limit Theorems for Dynamical Systems

“…The MLCLT is a generalization of the local central limit theorem (LCLT) and has been used in [45] and [17] to prove limit theorems for flows. The MLCLT has the following form…”

Local limit theorems for suspended semiflows

“…Versions of (Int-LLT) in the case where p = 2 (i.e. S is Gaussian) can be found in [37], [16] and [9].…”

“…whence by the weak convergence lemma µ f (x, t) dt < ∞ a.e., we can define (as in [37], [16] and [9]) the smooth cocycle…”