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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
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##### 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)
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