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On periodic homeomorphisms of spheres. (English) Zbl 1016.57019

The authors give natural lower estimates for metric data of orbits of a \(p\)-periodic homeomorphism \(h\) acting on the unit sphere \(S^n\) in the Euclidean \((n+1)\)-space. These are the shift \(\|h\|=\sup_{x\in S^n}\|h(x)- x\|\) and the maximal diameter \(\theta(h)\) of \(h\)-orbits. The bounds occurring herein are the following: \(\rho_p\) is the length of the side of a regular \(p\)-gon inscribed in the unit circle \(S^1\), \(d_p\) is the diameter of this \(p\)-gon, \(t_n\) is the edge length of the regular \((n+1)\)-simplex inscribed in \(S^n\).

MSC:

57S25 Groups acting on specific manifolds
52A40 Inequalities and extremum problems involving convexity in convex geometry
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
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References:

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