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

Coelho, Z.; Parry, W.; Williams, R.

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].

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

Keywords:

hyperbolic diffeomorphism; Livšic’s periodic point theorem; non-uniform hyperbolic attractor; Anosov diffeomorphism; automorphism of the first Čech homology group

Citations:

Zbl 0383.58010; Zbl 0211.26801