Livšic theorem for matrix cocycles. (English) Zbl 1238.37008

This important paper settles a long-standing open problem concerning matrix-valued cocycles over hyperbolic dynamical systems.
If \(f : X \to X\) is an invertible map, a cocycle over \(f\) with values in a group \(G\) is a map \(\mathcal{A} : X \times \mathbb{Z} \to G\) such that \(\mathcal{A}(x,0)\) is the identity of \(G\) and \[ \mathcal{A}(x,n + k) = \mathcal{A}(f^k x, n) \cdot \mathcal{A}(x,k) \] for all \(x \in X\) and \(n, k \in \mathbb{Z}\). If \(G = \text{GL}(m,\mathbb{R})\), \(\mathcal{A}\) is called a linear cocyle.
A cocycle is uniquely determined by its generator \(A : X \to G\), where \(A(x) = \mathcal{A}(x,1)\). Sometimes \(A\) itself is called a cocyle. A cocyle \(A\) is called a coboundary if there exists a map \(C : X \to G\) such that \(A(x) = C(fx) C(x)^{-1}\) for all \(x \in X\).
It is clear that any coboundary \(A\) must have trivial periodic data, i.e., for any periodic point \(p\) of \(f\) with period \(n\), \[ \mathcal{A}(p,n) = A(f^{n-1}p) \cdots A(fp) A(p) = \operatorname{Id}. \] The question is whether this necessary condition is also sufficient for \(A\) to be coboundary. This question was answered in the affirmative by A. N. Livšic in his seminal papers [“Homology properties of \(Y\)-systems”, Mat. Zametki 10, 555–564 (1971; Zbl 0227.58006); “Cohomology of dynamical systems”, Izv. Akad. Nauk SSSR, Ser. Mat. 36, 1296–1320 (1972; Zbl 0252.58007)] in the case when \(f\) is a hyperbolic dynamical system, \(G\) is an Abelian group, and \(A\) is Hölder continuous. The non-abelian case is much harder and despite much recent progress, many questions remain open.
The paper under review settles this problem for arbitrary linear Hölder cocycles over a hyperbolic dynamical system \(f\). More precisely (and generally), the main result of the paper states that if \(f\) is a topologically transitive homeomorphism of a compact metric space \(X\) having the so-called closing property (possessed by many important systems of interest) and if \(A : X \to \text{GL}(m,\mathbb{R})\) is an \(\alpha\)-Hölder cocycle with trivial periodic data, then there exists an \(\alpha\)-Hölder map \(C : X \to \text{GL}(m,\mathbb{R})\) such that \(A(x) = C(fx) C(x)^{-1}\) for all \(x \in X\).
In the more general case when the periodic data is not trivial but uniformly bounded, i.e., \(\mathcal{A}(p,n)\) lies in a compact set \(K\), for all \(n\)-periodic points \(p\) of \(f\), the author proves that there exists a compact set \(K'\) such that \(\mathcal{A}(x,n)\) lies in \(K'\) for all \(x \in X\) and \(n \in \mathbb{Z}\).
The proof of the main theorem relies on another result, important in its own right, which goes as follows. If \(f\) and \(\mathcal{A}\) are as above and \(\chi_{\text{min}}\) and \(\chi_{\text{max}}\) are numbers such that for every \(n\)-periodic point \(p\) and every eigenvalue \(\lambda\) of \(\mathcal{A}(p,n)\) one has \(\chi_{\text{min}} \leq \frac{1}{n} \log \lambda \leq \chi_{\text{max}}\), then for every \(\varepsilon > 0\) there exists a constant \(c_\varepsilon\) such that \[ \| \mathcal{A}(x,n) \| \leq c_\varepsilon \exp\{ (\chi_{\text{max}} + \varepsilon) n\} \] for all \(n \in \mathbb{N}\) and \(x \in X\), with an analogous result for \( \| \mathcal{A}(x,n)^{-1} \|\).
A further remarkable result used in the proof of the main theorem asserts that the Lyapunov exponents (with respect to an ergodic invariant measure) of a linear Hölder cocyle over \(f\) as above can be approximated by the Lyapunov exponents of \(\mathcal{A}\) at periodic points. Note the absence of any assumption on hyperbolicity of the cocycle.
The above-mentioned results hold more generally for any extension \(\mathcal{A}\) of \(f\) by linear transformations of a vector bundle over \(X\) if one can identify fibers at nearby points Hölder continuously via local trivialization of connection. In particular, this means that these results apply to the derivative cocycle of a smooth hyperbolic system as well as to its restriction to a Hölder continuous invariant distribution – without any global trivialization assumptions.
The paper is very well written and readable.


37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37H05 General theory of random and stochastic dynamical systems
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
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