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Invariance principles for diagonal flows on \(SL(d,\mathbb{R})/SL(d,\mathbb{Z})\). (Principes d’invariance pour les flots diagonaux sur \(SL(d,\mathbb{R})/SL(d,\mathbb{Z}))\).) (French) Zbl 1009.60018

Let \(G = SL(d;\mathbb{R}), \Gamma = SL(d;\mathbb{Z}), d \geqq 3,\) and \(X = G/\Gamma.\) Let \(\mu\) be the normalized measure on \(X\), invariant under left translation, induced from the Haar measure on \(G.\) Let \(T \in G\) be a diagonal matrix whose elements are positive, decreasing. Let \((T^{t}; t \in \mathbb{R})\) be a flow on \(X\) defined by given \(T \in G.\) Consider a dynamical system \((X,T,\mu)\). Let \(f \in L^{2}(X,d\mu)\) be such that its expectation \(E[f] = 0.\) Define its variance \(\sigma^{2}(f)\) as \(\sigma^{2}(f) = \lim_{n\rightarrow\infty}\frac{1}{n} \|\int_{0}^{n}f(T^{s}x)ds \|_{2}^{2}.\) Let us say a function \(f\) satisfies 1) the invariance principle of Donsker (PID) if the probability law of the stochastic process \(\xi_{t}(s,x) = \frac{1}{\sigma\sqrt{t}} \int_{0}^{st} f(T^{u}x)du, 0 \leq s \leq 1\), \(t \geq 0\), converges to the Wiener measure on \(C[0,1]\) as \(t \rightarrow \infty\), 2) the invariance principle of Strassen (PIS) if the set of \(\zeta_{t}(s,x) = \frac{1} {\sqrt{2t \sigma^{2}\log\log{t}}}\int_{0}^{st}f(T^{u}x) du\), \(0 \leq s \leq 1\), \(t \geq 0\), is relatively compact in \(C[0,1],\) a.e., and the set of its limiting points coincides with the set of absolutely continuous functions \(g\) on \([0,1]\) such that \(g(0) = 0\) and \(\int_{0}^{1} {g'}^{(t)^2} dt \leq 1.\) We call a function \(f\) on \(X\) is a co-boundary in the sense of the flow if there exists a mapping \(h\) on \(X\), finite, measurable such that for any \(t>0\) and a.a. \(x\), \(\int_{0}^{t}f(T^{s}x) ds = h(T^{t}x) - h(x).\) A function \(f\) defined on \(X\) is called (\(C,p\))-Hölder if \(|f(x) - f(y)|\leq Cd(x,y)^{p}\), where \(C > 0, p>0\) and \(d(x,y) \) is the distance of \(x,y \in X.\) For a set \(U \subset X,\) define \(\partial U(r), r>0,\) as \(\partial U(r)= \{x \in X \mid B_{0}(e,r)x \cap \partial U \neq\emptyset\},\) where \(B_{0}(e,r)\) is the ball in \(G\) with center \(e\) and radius \(r\). A function \(f\) defined on \(X\) is called \((C,p)\)-regular if \(\mu(\partial U(r)) \leq Cr^{p}\) for any \( r >0.\) The following two theorems are proved in the article.
Theorem A. Let \(f\) be a Hölder function or an idicator of a regular set. Then, if \(f\) is not a co-boundary, it satisfies PID and PIS.
Theorem B. A Hölder function which is a co-boundary is a co-boundary in the set of Hölder functions.
The definition of \(\xi_{n}(t,x)\) on p. 584 is insufficient. It must be \(\xi_{n}(k/n,x) = \frac{1}{\sigma \sqrt{n}} S_{k}\varphi(x)\), \(k = 1,\dots,n\).

MSC:

60F17 Functional limit theorems; invariance principles
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