Abstract
Let be an open subset and be an arbitrary local homeomorphism with . We compute the fixed point indices of the iterates of at , and we identify these indices in dynamical terms. Therefore, we obtain a sort of Poincaré index formula without differentiability assumptions. Our techniques apply equally to both orientation preserving and orientation reversing homeomorphisms. We present some new results, especially in the orientation reversing case.
Highlights
There is abundant literature about the fixed point index of a homeomorphism f, in a neighborhood of an isolated fixed point and the local dynamical behavior of f
It is well known that the classical Poincareindex formula relates the index of a planar vector field with the elliptic and hyperbolic regions in a neighborhood of a critical point
Let f : U ⊂ R2 → f U ⊂ R2 be a homeomorphism with p being an isolated fixed point of f, and let us assume that there is a strong filtration pair of period r, N, L, such that p ∈ int N \ L, L / ∅, f j cl N \ L ⊂ U for j ∈ {1, . . . , r} and Fix fr ∩ cl N \ L {p}
Summary
There is abundant literature about the fixed point index of a homeomorphism f, in a neighborhood of an isolated fixed point and the local dynamical behavior of f. The main technique in the proof of their theorem is the computation of the fixed point index of all iterates of an orientation preserving homeomorphism in a neighborhood of a fixed point p which is an isolated invariant set, neither an attractor nor a repeller. B Given any Jordan domain J, Inv cl J , f ∩ ∂ J / ∅ and an isolating block, N, is a neighborhood that isolates the fixed or periodical prime ends of the component of Inv cl J , f containing p, to prove that J and N determine canonically a number of generalized unstable stable branches and generalized repelling attracting petals around the fixed point see Definition 2.6.
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