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Fixed points of planar homeomorphisms of the form identity + contraction. (English) Zbl 1133.37012

Summary: Let \(f\) be a planar homeomorphism which has the form identity + contraction. We prove the existence of a fixed point of \(f\) under some geometrical condition on an orbit of \(f\). The paper improves the result of J. Aarao and M. Martelli [Topol. Methods Nonlinear Anal. 20, No. 1, 15–23 (2002; Zbl 1011.37021)] and provides an example which shows that, in the given setting, the theorem cannot be made stronger.

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37B30 Index theory for dynamical systems, Morse-Conley indices
55M25 Degree, winding number
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics

Citations:

Zbl 1011.37021
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References:

[1] J. Aarao and M. Martelli, Stationary states for discrete dynamical systems in the plane , Topological Methods Nonlinear Anal. 20 (2002), 15-23. · Zbl 1011.37021
[2] L.E. Brower, Beweiss des ebene translationsatzen , Math. Ann. 72 (1912), 37-54.
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[5] ——–, Homeomorphisms of two-dimensional manifolds , Houston J. Math. 11 (1985), 455-469. · Zbl 0605.57005
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[8] M. Martelli, Introduction to discrete dynamical systems and chaos , Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 1999. · Zbl 1127.37300
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