Lefschetz periodic point free self-maps of compact manifolds

Periods of Morse–Smale diffeomorphisms on $${\mathbb {S}}^n$$, $${\mathbb {S}}^m \times {\mathbb {S}}^n$$, $${\mathbb {C}}{\mathbf{P }}^n$$ and $${\mathbb {H}}{\mathbf{P }}^n$$

“…Remark 11. A map f is called Lefschetz periodic point free if L(f m ) = 0, for all m ∈ N. In [8], it is claimed that there are no Lefschetz periodic point free maps on CP n and HP n , that is, all self-maps of CP n and HP n have periodic points. Here, we see that in particular Morse-Smale diffeomorphisms in CP n and HP n always have fixed points, unless when n is odd and d = −1, where in this case, there are always fixed points or periodic points of period 2.…”

“…In [8], the authors study the Lefschetz periodic point free self-maps on S m × S n , CP n and HP n . Our results give extended information in the same line for the Morse-Smale diffeomorphisms.…”

On Lefschetz periodic point free self-maps

“…Note that if f is periodic point free, then f is Lefschetz periodic point free, but in general the converse does not hold. Periodic point free and Lefschetz periodic point free maps have been studied previously, see for instance [5,6,12,13].…”

“…Let f : X → X be a continuous self-map on X, its Lefschetz zeta function has the form: (5) ζ f (t) = (1 − t) −1 q 1 (t) (−1) n 1 +1 • • • q l (t) (−1) n l +1 , where q k (t) = det(Id * n k − tf * n k ), clearly it is a polynomial of degree smaller than or equal to s k , the degree is s k if and only if 0 is not an eigenvalue of f * n k . So if p k (t) is the characteristic polynomial of f * k , we have p k (t) = (−1) s k t s k q k (1/t).…”

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

“…Using the fact that α 2 = β 2 = 0 and well-known properties of cohomology ring, cf. [14] for details, we obtain * 4…”