Partially Periodic Point Free Self-Maps on Product of Spheres and Lie Groups
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Partially Periodic Point Free Self-Maps on Product of Spheres and Lie Groups Víctor F. Sirvent1 Received: 28 April 2020 / Accepted: 21 August 2020 © Springer Nature Switzerland AG 2020
Abstract Let X be a topological manifold and f : X → X a continuous map. We say the map f is partially periodic point free up to period n if f does not have periodic points of periods smaller than n + 1. A weaker notion is Lefschetz partially periodic point free up to period n, i.e. the Lefschetz numbers of the iterates of f up to n are all zero. Similarly f is Lefschetz periodic point free if the Lefschetz numbers of all iterates of f are zero. In the present article we consider continuous self-maps on products of any number spheres of different dimensions. We give necessary and sufficient conditions for such maps to be Lefschetz periodic point free and Lefschetz partially periodic point free. We apply these results to continuous self-maps on Lie groups. Keywords Lefschetz numbers · Periodic point · Product of spheres · Lie groups Mathematics Subject Classification 37C25 · 37E15 · 55M20
1 Introduction Let X be a topological space and f : X → X be a continuous map. We say that x ∈ X is a periodic point of period n if f n (x) = x and f i (x) = x for 1 ≤ i ≤ n − 1. We denote by Per(f) the set of all periods of f . We say that the map f is periodic point free, if f does not have periodic points, i.e. Per( f ) = ∅. The periodic point free maps on different spaces have been studied by several authors, for example on the 2–dimensional torus (cf. [2,13,19]) and on the annulus, see [12,17]. If Per( f ) ∩ {1, 2, . . . , n} = ∅ then we say that the map f is partially periodic point free up to period n. If n = 1, we say that f is fixed point free. See [5,28,30] for studies of different types of partially periodic point fee maps.
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Víctor F. Sirvent [email protected] Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile 0123456789().: V,-vol
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V.F. Sirvent
One of the most important techniques in studying the existence periodic points of continuous self-maps on manifolds, or more general on compact polyhedra, is the Lefschetz fixed point theory. Let N be the topological dimension of a compact manifold X . We denote by Hk (X , Q), for 0 ≤ k ≤ N , the homology linear spaces of X with coefficients over the rational numbers. These spaces are finite dimensional vector spaces over Q. Given a continuous map f : X → X , it induces linear maps f ∗k : Hk (X , Q) → Hk (X , Q), for 0 ≤ k ≤ N , all the entries of the matrices f ∗k are integer numbers. The Lefschetz number L( f ) is defined as L( f ) :=
N (−1)k trace( f ∗k ). k=0
The Lefschetz Fixed Point Theorem establishes the existence of a fixed point if L( f ) = 0 (cf. [20] or [4]). Clearly the converse is not always true, take for example the identity map on the torus or maps on the circle, of degree 1 (cf. [1]). In general it is not true that L( f m ) = 0 implies that f has a periodic point of period m; it only implies the existence of a peri
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