# zbMATH — the first resource for mathematics

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
Full Text: