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An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds. (English) Zbl 1364.37054

Summary: For a given self-map \(f\) of \(M\), a closed smooth connected and simply-connected manifold of dimension \(m\geq 4\), we provide an algorithm for estimating the values of the topological invariant \(D^m_r[f]\), which equals the minimal number of \(r\)-periodic points in the smooth homotopy class of \(f\). Our results are based on the combinatorial scheme for computing \(D^m_r[f]\) introduced by G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13, No. 1, 63–84 (2013; Zbl 1276.55005)]. An open-source implementation of the algorithm programmed in C++ is publicly available at http://www.pawelpilarczyk.com/combtop/.

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
55M20 Fixed points and coincidences in algebraic topology

Citations:

Zbl 1276.55005

Software:

CombTop
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