## Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2.(French)Zbl 0554.58034

Let $$X\neq \emptyset$$ be a metrisable compact space, $$g: X\to X$$ a continuous map, m a probability Radon measure on X, B a Banach algebra and $$A: X\to B$$ a continuous map. Let us denote $$A^ k_ g(x)=A(g^{k- 1}(x))...A(x)$$, $$k\geq 1$$, $$x\in X$$. The maximal Lyapunov exponent of the system (g,m,A) is defined by $\lambda_+(g,m,A)=\underline{\lim}_{k\to +\infty}(1/k)\int_{X}\log \| A^ k_ g(x)\| dm(x)\in [-\infty,+\infty).$ If (g,A) has a ”partial hyperbolic structure”, then $$\lambda_+(g,m,A)>0$$ for all m. The main aim of the paper is to give a general procedure to construct systems (g,m,A) such that (g,A) does not have ”partial hyperbolic structure”, but $$\lambda_+(g,m,A)>0$$. The main tool consists in the following theorem, which can be considered the abstract skeleton of the example from [the author, Ergodic Theory Dyn. Syst. 1, 65-76 (1981; Zbl 0469.58008)]:
Let $${\mathbb{D}}^ n$$ be the closed unit polydisc in $${\mathbb{C}}^ n$$, $${\mathbb{T}}^ n=\{z\in {\mathbb{C}}^ n;| z_ j| =1$$ for all $$j=1,...,n\}$$, m the Haar measure on $${\mathbb{T}}^ n$$ and f a $${\mathbb{C}}^ n$$-valued analytic map on a neighbourhood of $${\mathbb{D}}^ n$$, such that $$f({\mathbb{D}}^ n)\subset {\mathbb{D}}^ n$$, $$f({\mathbb{T}}^ n)\subset {\mathbb{T}}^ n$$, $$f(0)=0$$. Let further A be an analytic map from a neighbourhood of $${\mathbb{D}}^ n$$ in a complex Banach algebra B. Then $\lambda_+(f| {\mathbb{T}}^ n,m,A| {\mathbb{T}}^ n)\geq \log (spectral\quad radius\quad of\quad A(0)).$ A great number of significant examples are produced, some of them show also that the theorem of Arnold and Moser on $${\mathbb{T}}^ n$$ has for $$n\geq 2$$ a local character in contrast with the case $$n=1$$ [the author, Publ. Math., Inst. Hautes Etud. Sci. 49, 5-233 (1979; Zbl 0448.58019)].
We have to mention also that a rotation number function is defined and studied for homeomorphisms of the form $$X\times {\mathbb{T}}^ 1\ni (x,\theta)\mapsto (g(x),h(x)(\theta))\in X\times {\mathbb{T}}^ 1$$, where X is a metrisable compact space, $$g: X\to X$$ a homeomorphism and h a continuous map from X to the topological group of all orientation preserving homeomorphisms of $${\mathbb{T}}^ 1$$, such that h is homotopic to the constant map taking the value $$id_{{\mathbb{T}}^ 1}$$.
Reviewer: L.Zsido

### MSC:

 37A99 Ergodic theory

### Citations:

Zbl 0469.58008; Zbl 0448.58019
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