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**A note on Livšic’s periodic point theorem.**
*(English)*
Zbl 0897.58037

Nerurkar, M. G. (ed.) et al., Topological dynamics and applications. A volume in honor of Robert Ellis. Proceedings of a conference in honor of the retirement of Robert Ellis, Minneapolis, MN, USA, April 5–6, 1995. Providence, RI: American Mathematical Society. Contemp. Math. 215, 223-230 (1998).

Two examples of diffeomorphisms of the connected sum of two anchor rings are considered. The first is a hyperbolic diffeomorphism which induces an automorphism of the first homology group of the attractor having 1 as an eigenvalue. The second fails to be hyperbolic on two intersecting transverse disks and is non-uniformly hyperbolic on the rest of the attractor. The conclusion of Livšic’s periodic point theorem is not true for the later example. The construction of these diffeomorphisms is relevant to a problem first raised by M. Hirsch [Topology 10, 177-183 (1971; Zbl 0211.26801)] and mentioned by R. Bowen [‘On Axiom \(A\) diffeomorphisms’ (Regional Conf. Ser. in Math. 35, AMS, Providence) (1978; Zbl 0383.58010)]: is there an Anosov map which induces an automorphism of the first homology group having 1 as an eigenvalue? Both examples are modelled on maps of the one-dimensional branched (figure eight) manifold – the wedge of two circles. The presentation of diffeomorphisms are geometric and largely descriptive with details of the analysis only indicated.

For the entire collection see [Zbl 0882.00043].

For the entire collection see [Zbl 0882.00043].

Reviewer: E.Ershov (St.Peterburg)

### MSC:

37D99 | Dynamical systems with hyperbolic behavior |

37E99 | Low-dimensional dynamical systems |

28D05 | Measure-preserving transformations |

37A99 | Ergodic theory |

37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |