The least number of 2-periodic points of a smooth self-map of S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{S}^\mathbf{2}$$\end{document} of degree 2 equals 2

We show that there exists a smooth self-map of the sphere f:S2→S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:S^2\rightarrow S^2$$\end{document} which has degree 2 and has only two 2-periodic points.


Introduction
We fix a self-map f : M → M of a compact connected manifold and a natural number n. What is the least number of n-periodic points #Fix(h n ) where h runs through the homotopy class of f ? If we moreover restrict to simply connected M and we allow all continuous maps homotopic to f , then the least number is always 1 or 0 [4]. However, if h runs only through the smooth homotopy class of f , then the least number may be much larger, which was noticed by Shub and Sullivan [10]. This gave rise to D m n (f ) an algebraic lower bound of the number of n-periodic points in the smooth homotopy class of f , see [4]. In dimension m ≥ 3, the homotopy invariant D m n (f ) turned out to be the best lower bound, i.e., it can be realized by a smooth map homotopic to the given f (a Wecken-type theorem) [4] .
On the other hand, the sphere S 2 is the unique two-dimensional closed simply connected manifold. Surprisingly, the methods of reducing fixed and periodic points do not work in general on surfaces. The reason is that the Whitney trick of canceling intersection points does not hold in low dimensions, thus the Wecken theorem for periodic points works only from dimension 3 on. See [8,9]. This makes the problem of minimizing the number of periodic points open in dimension 2, in particular for self-maps of S 2 .
In [6], we started to study this case. In this paper and in [6] by smooth we mean C ∞ , since the maps in Lemma 2.2, which are given explicitly in the proof of Theorem 3.7 in [1], may by represented by C ∞ maps.
In Sect. 3 we give the algebraic necessary condition to homotope a selfmap of S 2 to a map with at most two n-periodic points. The main result of the paper is Theorem 3.2 saying that the above condition is also sufficient for deg(f ) = 2 and n = 2.
This must be done directly, since the techniques of reducing isolated periodic orbits with opposite indices, used in the Nielsen fixed and periodic point theory, are not available in dimension 2.

Indices of iterations of a smooth map
In 1983, Dold [3] noticed that a sequence of fixed point indices A k = ind(f k ; x 0 ), where f is a continuous self-map of a Euclidean space R m and x 0 is an isolated fixed point for each k, must satisfy some congruences. Namely for each n ∈ N k|n μ(n/k) · ind(f k ; x 0 ) ≡ 0(modulo n) where μ denotes the Möbius function.
It was shown [1] that each sequence of integers (A k ) satisfying Dold congruences can be realized as A k = ind(f k ; x 0 ), for a continuous self-map of R m for m ≥ 3. In other words, Dold congruences are the only restriction for a sequence of integers to realize the fixed point index of a continuous map.
Surprisingly, it turned out that there are much more restrictions on sequences A k = ind(f k ; x 0 ) when f is smooth [2,7,10]. In [5] it is proved that the necessary conditions given in [2] are, in dimension ≥ 3, also sufficient and the full description of all such sequences is given in [5]. We call them smoothly realizable in dimension m.
It is convenient to present the sequences of integers as the sum of the following elementary periodic sequences It is easy to notice that each integer sequence (A n ) can be written down uniquely in the following form of a periodic expansion: A n = ∞ l=1 a l reg l (n), where a n = 1 n l|n μ( n l ) A l for suitable a l ∈ R. Moreover, all coefficients a l are integers if and only if the sequence (A n ) satisfies Dold congruences.
The above observations, applied in dimension 2, resulted in the full description of all possible sequences smoothly realizable in dimension 2. In the next Lemma, we reformulate Theorem 3.7 in [1] using our notations Lemma 2.2. [1] see also Lemma 1.1 in [6].
Let U ⊂ R 2 be a neighborhood of (0, 0) and let f : U → R 2 be a smooth map such that (0, 0) is an isolated periodic point. Then, the periodic expansion of the local fixed point index of f takes one of the following forms: The right-hand side of the above formulae may be also written as (1) constant

Algebraic necessary condition
Let us fix a pair of numbers (d, n) ∈ Z × N. Does there exist a smooth map f : S 2 → S 2 satisfying deg(f ) = d and #Fix(f n ) ≤ 2? Theorem 1.1 allows to concentrate on |d| ≥ 2 and n ≥ 2. Suppose that there exists f a smooth map of degree d with at most two n-periodic points. We may assume that Fix(f n ) ⊂ {N Pole, SPole}. Then for some expressions C 1 , C 2 of types (1) − −(4) for all k|n. In other words, the existence of expressions C 1 , C 2 satisfying d k + 1 = C 1 (k) + C 2 (k) is a necessary algebraic condition to reduce Fix(f n ) to two points. Proof. ⇒. We assume that d k + 1 = C 1 (k) + C 2 (k) for k|n where C 1 , C 2 are of type (1) − (4). We will show that n is a prime. First we assume that one of C 1 , C 2 is of type (1). Then, the sum C 1 (k) + C 2 (k) takes at most two values.
Since d k + 1 takes distinct values (for a fixed |d| ≥ 2) and equality holds for all divisors of n, n must be a prime.
On the other hand if no of hence one of C i must be of type (4) and the other of type (3). Then C 1 + C 2 takes at most two values.
⇐. If p is a prime number then the equality d p is an integer by small Fermat's theorem.
In the rest of the paper, we will assume that n = d = 2. We will show that then the above condition is also sufficient. We will show that Theorem 3.2. There exists a smooth map f : S 2 → S 2 of degree 2 which has only two 2-periodic points.
This will be done as follows. First, we give a convenient formula of a map of degree 2. Then, we deform smoothly this map near the poles to realize their appropriate values of the fixed point index. Some extra 2-periodic points appear. The last step is to remove these points.

Remark 3.3.
To get a smooth map we start with a map which is smooth near the poles and we use continuous deformations which are constant near the poles. Finally we deform the obtained map, with only two 2-periodic points, to a smooth map by a homotopy constant near the poles. If the last deformation is sufficiently small, the poles remain the unique 2-periodic points.

Notation
Let us introduce some notation. We will consider the sphere S 2 as the quotient and φ denotes a real number modulo 2π. We will refer to θ ,φ as latitude and longitude, respectively. We also denote by three sectors of the sphere. We will denote N Pole = [ π 2 , * ] and SPole = [− π 2 , * ]. Please notice that symbols S 2 + , S 2 − have a different meaning in [6].
Vol. 21 (2019) The least number of 2-periodic points of a smooth Page 5 of 13 14 We define a continuous map f 0 : The degree of f 0 equals 2, since the restrictions of f 0 to S 2 − and S 2 + are orientation-preserving diffeomorphisms . Proof. We notice that f 0 has exactly one fixed point in each sector: N Pole ∈ S 2 + , [0, 0] ∈ S 2 0 , SPole ∈ S 2 − , since the coordinate θ is being expanded on each sector. Now we look for 2-orbits. The above argument (θ is expanding) implies that there is no 2-orbit contained in a sector. Moreover the component φ excludes a 2-orbit with a point in S 2 + and the other one in S 2 − . Now each 2-orbit must have one element in S 2 0 and the second in

Removing the orbit {b, f (b)}
In this section, we will remove the orbit {b, f (b)} and the fixed point [0, 0] by a deformation with the carrier in a neighborhood of the arc < b, f(b) >.
Here is the sketch of the deformation. We start by a smooth deformation of f 0 near SPole adding an additional fixed point b . The orbit {b, f 0 (b)}, the points [0, 0] and b belong to the arc < SPole, [π/6, 0] >. The map f 0 sends the ends of the last arc to SPole, the middle point of the arc goes to N Pole and the f 0 is linear on each half of the arc. The restrictions of f 0 to neighbor arcs < SPole, [π/6, φ 0 ] > look similar. We consider the region for −π/2 ≤ θ ≤ −π/6 and φ = 0 for −π/6 ≤ θ ≤ π/6. We deform the restriction of f 0 to the arc < SPole, [π/6, 0] >, keeping the end points fixed, by squeezing the arc to a neighborhood of SPole so that there is no periodic point inside the arc. Then, we extend this deformation to V 1 by a homotopy which keeps the boundary bdV 1 fixed, the meridians are sent into themselves, or to the meridian 0. Finally, we compose the obtained deformation with a homeomorphism of V 1 which is constant on the boundary and for φ = 0 and which makes |φ| smaller elsewhere. The final map is a self-map of degree 2 with no periodic points inside V 1 . This gives the maps Now we go to the details. We start with a deformation of f 0 near SPole. We introduce polar coordinates in S 2 \N Pole by identifying Since the map f 0 near the South Pole has the form f 0 [θ, φ] = [3θ + π, φ], in the new coordinates we get the mapf (z) = 3z (for |z| < π 3 ). Lemma 7.1, Remark 7.3 and the above coordinates give a deformation of f 0 to a map which we will denote by f 1  (−π/2, −π/2) (π/6, −π/2) (−π/2, π/2) (π/6, π/2) We start by analyzing the restriction f 1| : [−π/2, π/6]×0 → [−π/2, π/2] ×0. The graph is given in Fig. 1, where the asymmetry in the left lower corner is resulted by the above deformation of f 0 to f 1 . Let (b , 0) be the unique fixed point in (−π/2, 0) × 0. We fix a point b − ∈ (−π/2, b ) and we denote C = f 1 (b − ). Then −π/2 < C < b − and we fix another point b ∈ (C, b ) (see the bottom of Fig. 1). Now the formulā f 1 (θ) = min(p 1 f 1 (θ, 0); C) contracts interval [−π/2, π/6] near SPole. Here p 1 , p 2 denote the projections of [−π/2; +π/2] × [− ; + ]. Now we define the map f 2| : Roughly speaking: [b − ; b + ] × 0 is squeezed near SPole and R makes |φ| smaller. Since f 2| coincides with f 1 on the boundary, we may define the map We check that f 2 satisfies (1) − (3). (1), (2) follow straight from the definition. We check (3).

Removing the orbit a, f (a)
In this section, we deform the map f 0 (not f 2 ) and we remove the other 2-orbit {a, f (a)}. The carrier of the deformation will be disjointed from the carrier of the previous deformation. At the end of the section, we will show that the two deformations give a map of degree 2 whose 2-periodic points are only N Pole and SPole. This will end the proof of Theorem 3. 2 We will say that a subset A ⊂ S 2 is S 2 -convex if A does not contain antipodal points and for each a, a ∈ A the geodesic joining the points is contained in A.
We will deform the map f 0 only in the northern hemisphere, hence we introduce other polar coordinates Let f 3 be the induced map of S 2 . We will cancel simultaneously the orbits; {a = [ 3π 10 ; 0]; f 0 (a) = [− π 10 ; π]} and {w 0 , w 2 } by a homotopy with the carrier in an arbitrarily prescribed neighborhood of the arc < f 3 (a); w 2 >⊂ S 2 .
We define K 1 as the time-1 map of the vector field K.
Proof of Lemma 7.1. The first two properties follow from Lemma 7.2, since the map Φ 1 preserves sectors S k and half-lines L k . To prove the third property we notice that the vectors φ(z) , az are collinear ⇐⇒ z ∈ L k for a k = 0, . . . , 2n − 1. Moreover they have the same direction for k odd and are opposite for k even. Now, for k odd, their convex combination never vanishes K(z) = 0. Similarly, for k even, K has exactly one zero for 1 ≤ |z| ≤ 2 , z ∈ L k , since η is nonincreasing. Now we prove (5). We notice that in each fixed point z 2k the map K 1 is expanding the line L 2k and is squeezing at the orthogonal direction (since so does Φ 1 ). Now fixed point index equals (−1) · (+1) = −1. The same argument works for all iterations of K 1 . To prove (6) we notice that the total index must be +1, hence ind(K m 1 ; (0, 0)) = 1 − n(−1) = 1 + n for any m ∈ N. Similarly for n = 2 we get the Fig. 6 and a map K 1 with two additional fixed points each of index −1.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4. 0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.