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Let $f\colon M\to M$ be a self-map of a compact manifold and $n\in \mathbb{N}$. The least number of $n$-periodic points in the smooth homotopy class of $f$ may be smaller than in the continuous homotopy class. We ask: for which self-maps $f\colon M\to M$ the two minima are the same, for each prescribed multiplicity? In the study of self-maps of tori and compact Lie groups a necessary condition appears. Here we give a criterion which helps to decide whether the necessary condition is also sufficient. We apply this result to show that for self-maps of the product of the lens space $M=L(3)\times L(3)$ the necessary condition is also sufficient.

Let $$f:M\rightarrow M$$ f : M → M be a self-map of a compact manifold and $$n\in {\mathbb {N}}$$ n ∈ N . In general, the least number of n-periodic points in the smooth homotopy class of f may be much bigger than in the continuous homotopy class. For a class of spaces, including compact Lie groups, a necessary condition for the equality of the above two numbers, for each iteration $$f^n$$ f n , appears. Here we give the explicit form of the graph of orbits of Reidemeister classes $$\mathcal {GOR}(f^*)$$ GOR ( f ∗ ) for self-maps of projective unitary group PSU(2) and of $$PSU(2)\times PSU(2)$$ P S U ( 2 ) × P S U ( 2 ) satisfying the necessary condition. The structure of the graphs implies that for self-maps of the above spaces the necessary condition is also sufficient for the smooth minimal realization of n-periodic points for all iterations.

There are two algebraic lower bounds of the number of n-periodic points of a self-map f:M\to M of a compact smooth manifold of dimension at least 3 : NF_n(f)=min {#Fix(g^n) ;g\sim f; g continuous} and NJD_n(f)=min {#Fix}(g^n) ;g\sim f; g smooth}}. In general NJD_n(f) may be much greater than NF_n(f). We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism, the equality NF_n(f)=NJD_n(f) holds for all n iff all eigenvalues of a quotient cohomology homomorphism induced by f have moduli \le 1., accepted in SCIENCE CHINA Mathematics

We show for which (d, n) ∈ ℤ × ℕ there exists a smooth self-map f: S2 → S2 so that deg(f) = d and Fix(fn) is a point.

Let M be a smooth compact and simply-connected manifold with simply-connected boundary $$\partial M$$ , r be a fixed odd natural number. We consider f, a $$C^1$$ self-map of M, preserving $$\partial M$$ . Under the assumption that the dimension of M is at least 4, we define an invariant $$D_r(f;M,\partial M)$$ that is equal to the minimal number of r-periodic points for all maps preserving $$\partial M$$ and $$C^1$$ -homotopic to f. As an application, we give necessary and sufficient conditions for a reduction of a set of r-periodic points to one point in the $$C^1$$ -homotopy class.

There are two algebraic lower bounds of the number of n-periodic points of a self-map \({f : M \rightarrow M}\) of a compact smooth manifold of dimension at least 3: $$NF_{n}(f) = min\{\#Fix(g^{n}); g ~ f; g continuous\}$$ and $$NJD_{n}(f) = min\{\#Fix(g^{n}); g ~ f; g smooth\}.$$ In general NJDn(f) may be much greater than NFn(f). In the simply connected case, the equality of the two numbers is equivalent to the sequence of Lefschetz numbers satisfying restrictions introduced by Chow, Mallet-Parret and Yorke (1983). The last condition is not sufficient in the non-simply connected case. Here we give some conditions which guarantee the equality when \({\pi_{1}M = \mathbb{Z}_{2}}\).

Let $r$ be an odd natural number, $M$ a compact simply-connected smooth manifold, $\dim M\geq 4$, such that its boundary $\partial M$ is also simply-connected. We consider $f$, a $C^1$ self-maps of $M$, preserving $\partial M$. In [G. Graff and J. Jezierski, Geom. Dedicata 187 (2017), 241-258] the smooth Nielsen type periodic number $D_r(f;M,\partial M)$ was defined and proved to be equal to the minimal number of $r$-periodic points for all maps preserving $\partial M$ and $C^1$-homotopic to $f$. In this paper we demonstrate a purely combinatorial method of calculation of the invariant and illustrate it in various cases.

We show that there exists a smooth self-map of the sphere $$f:S^2\rightarrow S^2$$ which has degree 2 and has only two 2-periodic points.

We give an algebraic proof of the Theorem of Cheng Ye You that the least number of $n$-periodic points, in the continuous homotopy class of a self-map of a~torus, can be realized by a smooth map.