2008/05/08 by Alexander Blokh, Blokh, Alexander, Lex Oversteegen +1 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals #math.GN #msc:37B45 #msc:37C25 #msc:37F10 #msc:37F50 #msc:54F15 #msc:54H25
paper · pdf · doi:10.48550/arxiv.0805.1069
21 pages with corrected references
arxiv created 2008/09/21 · arxiv updated 2016/01/25
If f:[a,b]→ ℝ, with a<b, is continuous and such that a and b are mapped in opposite directions by f, then f has a fixed point in I. Suppose that f:ℂ→ℂ is map and X is a continuum. We extend the above for certain continuous maps of dendrites X→ D, X⊂ D and for positively oriented maps f:X→ ℂ, X⊂ ℂ with the continuum X not necessarily invariant. Then we show that in certain cases a holomorphic map f:ℂ→ℂ must have a fixed point a in a continuum X so that either a∈ Int(X) or f exhibits rotation at a.