Global shadowing of pseudo-Anosov homeomorphisms. (English) Zbl 0576.58025

Handel, Michael

Ergodic Theory Dyn. Syst. 5, 373-377 (1985).

J. Franks [Global analysis, Proc. Symp. Pure Math. 14, 61-93 (1970; Zbl 0207.543)] proved that a map which is homotopic to an Anosov diffeomorphism on a compact manifold is semi-conjugate, by a map homotopic to the identity, to the Anosov diffeomorphism. The main result in the present paper can be considered as generalization of this fact to pseudo-Anosov homeomorphisms on closed surfaces. More precisely, if \(f: M^ 2\to M^ 2\) is a pseudo-Anosov homeomorphism and \(g: M^ 2\to M^ 2\) is a map homotopic to f there is a closed subset \(Y\subseteq M^ 2\) and a surjection \(\phi\) : \(Y\to M\) homotopic to the inclusion such that \(f\circ \phi =\phi \circ g| Y.\)
The proof of this result uses the concept of K-global shadowing which is itself a generalization of \(\epsilon\)-shadowing as defined by R. Bowen [On axiom A diffeomorphisms (1978; Zbl 0383.58010)].

Reviewer: C.Chicone

MSC:

37D99

Dynamical systems with hyperbolic behavior

Keywords:

Anosov diffeomorphism; pseudo-Anosov homeomorphisms; shadowing

References:

[1]	Franks, Proceedings of the Symposium in Pure Mathematics 14 pp 61– (none) · doi:10.1090/pspum/014/0271990
[2]	Bowen, Regional conference series in mathematics 35 pp none– (1978)
[3]	Brown, The Lefschetz Fixed Point Theorem (1971) · Zbl 0216.19601
[4]	Fathi, Asterisque none pp 66– (1979)