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**Toward a theory of rank one attractors.**
*(English)*
Zbl 1181.37049

This paper extends some of the results of a previous paper by the same authors [Commun. Math. Phys. 218, No. 1, 1–97 (2001; Zbl 0996.37040)]. In that paper, the authors presented a general theory of strange attractors with one unstable direction and one strongly contracting direction. The results obtained were based on the methods in [M. Benedicks and L. Carleson, Ann. Math. (2) 133, No. 1, 73–169 (1991; Zbl 0724.58042)] by replacing the equation of the Hénon maps with geometric conditions.

The paper reviewed here considers strange attractors with one unstable direction and \(m-1\) strongly contracting directions (rank one attractors), where \(m\) is the dimension of the phase space. It is shown for large classes of one parameter families of maps \(\{T_a\}\) that \(T_a\) has a rank one attractor and admits an SRB measure for a positive measure set of parameters \(a\).

In Part I, necessary results for one dimensional maps are presented especially for Misiurewicz maps [see M. Misiurewicz, Publ. Math., Inst. Hautes Étud. Sci. 53, 17–51 (1981; Zbl 0477.58020)] and their perturbations. In Part II, a vital class of maps \({\mathcal G}\) with rank one attractors is constructed. Maps in \({\mathcal G}\) are nonuniformly hyperbolic and admit SRB measures. In Part III, the existence of maps in \({\mathcal G}\) is shown and the Main Theorem is proven.

This paper is self-contained and many of the proofs appear in the Appendix to maintain the flow of the paper. Lastly, the authors mention in the Introduction that further results for the maps in \({\mathcal G}\) will be published separately.

The paper reviewed here considers strange attractors with one unstable direction and \(m-1\) strongly contracting directions (rank one attractors), where \(m\) is the dimension of the phase space. It is shown for large classes of one parameter families of maps \(\{T_a\}\) that \(T_a\) has a rank one attractor and admits an SRB measure for a positive measure set of parameters \(a\).

In Part I, necessary results for one dimensional maps are presented especially for Misiurewicz maps [see M. Misiurewicz, Publ. Math., Inst. Hautes Étud. Sci. 53, 17–51 (1981; Zbl 0477.58020)] and their perturbations. In Part II, a vital class of maps \({\mathcal G}\) with rank one attractors is constructed. Maps in \({\mathcal G}\) are nonuniformly hyperbolic and admit SRB measures. In Part III, the existence of maps in \({\mathcal G}\) is shown and the Main Theorem is proven.

This paper is self-contained and many of the proofs appear in the Appendix to maintain the flow of the paper. Lastly, the authors mention in the Introduction that further results for the maps in \({\mathcal G}\) will be published separately.

Reviewer: Steve Pederson (Atlanta)

### MSC:

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

37E05 | Dynamical systems involving maps of the interval |

37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |

37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |