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Temporal distributional limit theorems for dynamical systems. (English) Zbl 1368.37016
The authors consider a Borel flow $$(T^s)$$ and a Borel function $$f$$ on a compact manifold $$X$$, the ergodic sums $$S_n(x):=\sum\limits^{n-1}_{k=0}f(T^kx)$$ or integrals $$J_t(x):=\int^t_0f(T^sx)ds$$, and the following five different kinds of distributional limit theorems (DLT): = 0.6 cm
(1)
The ergodic sums $$S_n(x)$$ satisfy a spatial DLT on a probability space $$(X,\mu)$$ when there are real constants $$B_n\to\infty$$ and $$A_n$$ such that the law of $$x\mapsto\frac{S_n(x)-A_n}{B_n}$$ under $$\mu$$ converges as $$n\to\infty$$ (towards some non-Dirac law).
(2)
The ergodic sums $$S_n(x)$$ satisfy a temporal DLT on the orbit of $$x$$ when there are real sequences $$B_N(x)\to\infty$$ and $$A_N(x)$$ such that the law of $$n\mapsto\frac{S_n(x)-A_N(x)}{B_N(x)}$$ under the uniform law $$\mathcal U\{0,\dots,N\}$$ converges as $$N\to\infty$$ (towards some non-Dirac law).
(3)
The ergodic sums $$S_n(x)$$ satisfy a strong temporal DLT when they satisfy a temporal DLT on the orbit of $$\mu$$-almost every $$x$$ and the normalizing sequence $$B_N$$ can be chosen to be independent of $$x$$.
(4)
The ergodic sums $$S_n(x)$$ satisfy an almost sure DLT on a probability space $$(X,\mu)$$ when there are real constants $$B_N\to\infty$$ and $$A_N$$ such that for $$\mu$$-almost every $$x$$ the law of $$n\mapsto\frac{S_n(x)-A_N}{B_N}$$, under the law on $$\{1,\dots,N\}$$ prescribing a mass $$\frac{c}{n}$$ at $$n$$, converges as $$N\to\infty$$ (towards some non-Dirac law).
(5)
The ergodic sums $$S_n(x)$$ satisfy a spatio-temporal DLT on a probability space $$(X,\mu)$$ when there are real constants $$B_n\to\infty$$ and $$A_ n$$ such that the law of $$(n,x)\mapsto\frac{S_n(x)-A_n}{B_n}$$ under $$\mathcal U\{0,\dots,N\}\otimes\mu$$ converges as $$n\to\infty$$ (towards some non-Dirac law).
The analogous notions (and results as well) regarding the ergodic integrals $$J_t(x)$$ are very similar. The main aim of the authors is to specify, in three classical cases, which among the spatial, strong temporal, almost sure and spatio-temporal DLT hold and which do not. A consequence will be that a large diversity of possible scenarios emerges, so that generally none of theses notions implies another one. The three classical cases which the authors analyze are Anosov flows, irrational rotations on the circle and horocycle flows of a Riemannian surface.
Some results presented in this interesting article are adapted from preceding ones, by a series of different authors, some are new extensions of previously known theorems, and some are new.
It is proved that for Anosov flows, the spatial, almost sure temporal and spatio-temporal DLT hold while the strong temporal DLT does not; for irrational rotations on the circle, the strong temporal and spatio-temporal DLT hold while the spatial and almost sure temporal DLT do not; for horocycle flows, depending on the class of Sobolev regularity of $$f$$, a series of various answers emerge, from which apparently no generality derives.

##### MSC:
 37A50 Dynamical systems and their relations with probability theory and stochastic processes 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 60F05 Central limit and other weak theorems 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37E10 Dynamical systems involving maps of the circle 37E35 Flows on surfaces 11K06 General theory of distribution modulo $$1$$ 37A17 Homogeneous flows 37C55 Periodic and quasi-periodic flows and diffeomorphisms
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