×

zbMATH — the first resource for mathematics

Global shadowing of pseudo-Anosov homeomorphisms. (English) Zbl 0576.58025
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
PDF BibTeX XML Cite
Full Text: DOI
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)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.