##
**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.

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.

Reviewer: Slobodan Simic (Berkeley)

### MSC:

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 |

### References:

[1] | L. Barreira and Y. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Cambridge: Cambridge Univ. Press, 2007, vol. 115. · Zbl 1144.37002 |

[2] | E. R. Goetze and R. J. Spatzier, ”On Liv\vsic’s theorem, superrigidity, and Anosov actions of semisimple Lie groups,” Duke Math. J., vol. 88, iss. 1, pp. 1-27, 1997. · Zbl 0879.22004 · doi:10.1215/S0012-7094-97-08801-3 |

[3] | B. Kalinin and V. Sadovskaya, ”Linear cocycles over hyperbolic systems and criteria of conformality,” J. Mod. Dyn., pp. 419-441, 2010. · Zbl 1225.37062 · doi:10.3934/jmd.2010.4.419 |

[4] | A. Katok, ”Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,” Inst. Hautes Études Sci. Publ. Math., iss. 51, pp. 137-173, 1980. · Zbl 0445.58015 · doi:10.1007/BF02684777 |

[5] | A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge: Cambridge Univ. Press, 1995, vol. 54. · Zbl 0878.58020 |

[6] | A. Katok and V. Nicticua, Differentiable Rigidity of Higher Rank Abelian Group Actions. |

[7] | R. de la Llave, ”Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems,” Comm. Math. Phys., vol. 150, iss. 2, pp. 289-320, 1992. · Zbl 0770.58029 · doi:10.1007/BF02096662 |

[8] | R. de la Llave, J. M. Marco, and R. Moriyón, ”Canonical perturbation theory of Anosov systems and regularity results for the Liv\vsic cohomology equation,” Ann. of Math., vol. 123, iss. 3, pp. 537-611, 1986. · Zbl 0603.58016 · doi:10.2307/1971334 |

[9] | R. de la Llave and A. Windsor, ”Liv\vsic theorems for non-commutative groups including groups of diffeomorphisms, and invariant geometric structures,” Ergodic Theory Dynam. Systems, vol. 30, pp. 1055-1100, 2010. · Zbl 1230.37042 · doi:10.1017/S014338570900039X |

[10] | A. N. Livvsic, ”Certain properties of the homology of \(Y\)-systems,” Mat. Zametki, vol. 10, pp. 555-564, 1971. |

[11] | A. N. Livvsic, ”Cohomology of dynamical systems,” Math. USSR Izvestija, vol. 6, pp. 1278-1301, 1972. · Zbl 0273.58013 · doi:10.1070/IM1972v006n06ABEH001919 |

[12] | M. Nicol and M. Pollicott, ”Liv\vsic’s theorem for semisimple Lie groups,” Ergodic Theory Dynam. Systems, vol. 21, iss. 5, pp. 1501-1509, 2001. · Zbl 0999.37017 · doi:10.1017/S0143385701001729 |

[13] | V. Nicticua and A. Török, ”Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices,” Duke Math. J., vol. 79, iss. 3, pp. 751-810, 1995. · Zbl 0849.58049 · doi:10.1215/S0012-7094-95-07920-4 |

[14] | V. Nicticua and A. Török, ”Regularity of the transfer map for cohomologous cocycles,” Ergodic Theory Dynam. Systems, vol. 18, iss. 5, pp. 1187-1209, 1998. · Zbl 0918.58057 · doi:10.1017/S0143385798117480 |

[15] | W. Parry, ”The Liv\vsic periodic point theorem for non-abelian cocycles,” Ergodic Theory Dynam. Systems, vol. 19, iss. 3, pp. 687-701, 1999. · Zbl 0988.37006 · doi:10.1017/S0143385799146789 |

[16] | W. Parry and M. Pollicott, ”The Liv\vsic cocycle equation for compact Lie group extensions of hyperbolic systems,” J. London Math. Soc., vol. 56, iss. 2, pp. 405-416, 1997. · Zbl 0943.37009 · doi:10.1112/S0024610797005474 |

[17] | M. Pollicott and C. P. Walkden, ”Liv\vsic theorems for connected Lie groups,” Trans. Amer. Math. Soc., vol. 353, iss. 7, pp. 2879-2895, 2001. · Zbl 0986.37020 · doi:10.1090/S0002-9947-01-02708-8 |

[18] | F. Rodriguez Hertz, ”Global rigidity of certain abelian actions by toral automorphisms,” J. Mod. Dyn., vol. 1, iss. 3, pp. 425-442, 2007. · Zbl 1130.37013 · doi:10.3934/jmd.2007.1.425 |

[19] | K. Schmidt, ”Remarks on Liv\vsic’ theory for nonabelian cocycles,” Ergodic Theory Dynam. Systems, vol. 19, iss. 3, pp. 703-721, 1999. · Zbl 0955.37018 · doi:10.1017/S0143385799146790 |

[20] | S. J. Schreiber, ”On growth rates of subadditive functions for semiflows,” J. Differential Equations, vol. 148, iss. 2, pp. 334-350, 1998. · Zbl 0940.37007 · doi:10.1006/jdeq.1998.3471 |

[21] | C. P. Walkden, ”Liv\vsic regularity theorems for twisted cocycle equations over hyperbolic systems,” J. London Math. Soc., vol. 61, iss. 1, pp. 286-300, 2000. · Zbl 0943.37010 · doi:10.1112/S0024610799008418 |

[22] | Z. Wang and W. Sun, ”Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits,” Trans. Amer. Math. Soc., vol. 362, iss. 8, pp. 4267-4282, 2010. · Zbl 1201.37025 · doi:10.1090/S0002-9947-10-04947-0 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.