Journal of Nonlinear Modeling and Analysis http://jnma.ca; http://jnma.ijournal.cn Volume 4, Number 4, December 2022, 1–11 DOI:10.12150/jnma.2022.1 Polynomial Homogeneous Maps and Their Periods∗ Jaume Llibre1 and Violetta Pilyugina1 Abstract We study the set of periods of the homogeneous polynomial maps f : Rn → Rn and f : Cn → Cn of degree m > 1. For these complex maps, we also describe the number of invariant straight lines through the origin by f k for k = 1, 2, . . . and the dynamics of f k over them. Keywords Homogeneous polynomial map, Period, Set of periods, Self–maps of a sphere. MSC(2010) 37F10, 37C25. 1. Introduction and statement of the main results We consider discrete dynamical systems given by a real or complex homogeneous polynomial map defined in Rn or Cn respectively. For the discrete dynamical sys- tems the periodic orbits play an important role for understanding their dynamics. Perhaps the best known example in this direction are the results contained in the paper entitle “Period three implies chaos” for continuous self–maps on the interval (see [21] or the book [2]). The real homogeneous polynomial maps f : Rn → Rn have been studied by many authors (see for instance the survey of Aliashvili [1] and the references quoted therein). However, not too much attention has been put in the study of their periodic orbits with the exception of their fixed points (see for instance [8, 19, 27]). Let CPn be the complex projective space of dimension n. The complex homo- geneous polynomial maps f : Cn → Cn also has been considered in [1] and in the references quoted there, but again not too much attention was put into their pe- riodic orbits. On the other hand for the complex homogeneous polynomial maps f : CPn → CPn their set of periods have been studied (see Fornaes and Sibony [11], or [10]). On the other hand, these maps also have been studied from the point of view of their degree (see [17]). The study of the set of periods of the real homogeneous polynomial maps f : Rn → Rn is our objective for such maps, while for complex homogeneous polynomial maps f : Cn → Cn our main goal is to the study their invariant straight lines † the corresponding author. Email address: jllibre@mat.uab.cat (J. Llibre), violetta.piliugina@gmail.com (V. Pilyugina) 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain ∗ The first author is partially supported by the Agencia Estatal de Investi- gación (Grant No. PID2019-104658GB-I00) and the H2020 European Re- search Council (Grant No. MSCA-RISE-2017-777911). 2 J. Llibre & V. Pilyugina through the origin of coordinates by f k for k = 1, 2, . . ., and the dynamics of the map f k restricted to these straight lines. Let F be the set of all real R or complex C numbers, and let F[x1 , . . ., xn ] be the ring of all polynomials in the n variables x1 , . . . , xn with coefficients in F. A polynomial P (x1 , . . . , xn ) of degree m is homogeneous, if P (λx1 , . . . , λxn ) = λm (x1 , . . . , xn ) for all λ ∈ F \ {0}. A homogeneous polynomial map f : Fn → Fn of degree m is a map f = (f1 , . . . , fn ), where fi ∈ F[x1 , . . . , xn ] is a homogeneous polynomial of degree m for all i = 1, . . . , n. Here, f k denotes the composition of the map f with itself k times. A point x ∈ Fn is fixed by the map f , if f (x) = x. Let k > 1 be a positive integer. A point x ∈ Fn is k–periodic or periodic of period k by the map f , if f k (x) = x and f j (x) 6= x for j = 1, . . . , k − 1. The set {x, f (x), f 2 (x), . . . , f k−1 (x)} is the periodic orbit of x. We say that the fixed points of f have period 1. We shall denote by Per(f ) the set of periods of all periodic points of the map f . Clearly Per(f ) is a subset of the set N of all positive integers. The sets Per(f ) when f is a homogeneous polynomial map of degree m change completely, if the homogeneous polynomial map is real or complex. If the degree m = 1, then the map is linear and its dynamics is easy to study. Here, we only consider homogeneous polynomial maps of degree m > 1. In order to state our result for the complex homogeneous polynomial maps of degree m > 1, we need some preliminary notions. Let f : Cn → Cn be a homogeneous polynomial map of degree m. For each x ∈ Cn \ {0}, we define the straight line Lx through the origin of Cn as Lx = {µx : for all µ ∈ C}, and we say that x is a director vector of Lx . A straight line through the origin of Cn with director vector x is invariant by f , if f (x) = λx for some λ ∈ C \ {0}. Then, f (Lx ) = Lx . It is clear that every k–periodic point x of f is on the invariant straight line Lx of f k , i.e., f k (Lx ) = Lx . Moreover, the set of straight lines {Lx , Lf (x) , Lf 2 (x) , . . . , Lf k−1 (x) } is a k–periodic orbit of f in the set of all invariant straight lines through the origin, i.e., f k (Lx ) = Lx and f ` (Lx ) 6= Lx for ` = 1, . . . , k − 1. The Möbius function µ : N → N is defined as   1, if r = 1,    µ(r) = 0, if some k 2 |r for some k ∈ N,    (−1)s , if r = p · · · p (distinct primes).  1 s Theorem 1.1. Let f : Cn → Cn be a homogeneous polynomial map of degree m > 1 such that f (x) 6= 0, if x 6= 0. Let k be a positive integer. Periods of Polynomial Homogeneous Maps 3 (a) The number of invariant straight lines through the origin for the map f k is (mkn − 1)/(mk − 1) taking into account their multiplicities. (b) The number of `–periodic invariant straight lines through the origin for the map f k is X mk`n/r − 1 µ(r) k`/r , m −1 r|` where µ : N → N is the Möbius function, and if all the multiplicities of the invariant straight lines of statement (a) are simple. (c) If Lx is an invariant straight line of f k , then Per(f k |Lx ) = N. (d) All the periodic points of the map f k |Lx with the exception of the origin are on a circle centered at the origin of coordinates, and these periodic points are repelling. In fact, this circle is the Julia set of the map f k |Lx : C → C, where we have identified Lx with C. (e) The number of `–periodic points of the map f k |Lx is X µ(r)mk`/r , r|` where µ : N → N is the Möbius function. The multiplicities mentioned in Theorem 1.1 are in the sense of the multiplicities of the Bezout Theorem (see for more details [7,12,26]). For a definition of the Julia set, see for instance [9]. Theorem 1.1 is proved in section 2. By Theorem 1.1, every homogeneous polynomial map f : Cn → Cn of degree m > 1 has k–periodic points for all positive integer k. Hence, from the point of view of the periodicity, these maps are not very interesting. The situation is completely different from homogeneous polynomial map f : Rn → Rn . The homogeneous polynomial maps f : CPn → CPn of degree d have been studied by Fornaes and Sibony [11]. Their results are related with the ones presented in statement (a) Theorem 1.1 for the homogeneous polynomial maps f : Cn+1 → Cn+1 of degree m, because the periodic points of the homogeneous polynomial maps of CPn are the periodic straight lines through the origin of Cn+1 (see for more details Corollary 3.2 of [11], and see also [23] and [25]). We denote the Euclidean norm of x ∈ Rn by ||x||. Let Sn−1 be the unit sphere of Rn , i.e., Sn−1 = {x ∈ Rn : ||x|| = 1}. Let f : Rn → Rn be a homogeneous polynomial map of degree m such that f (x) 6= 0, if x 6= 0. Then, the map f (x) F = Ff : Sn−1 → Sn−1 is defined by F (x) = . ||f (x)|| Let Z be the set of all integer numbers. The number of elements of the set ( n ) n X 1 (x1 , . . . , xn ) ∈ Z : 0 ≤ xi < m, xi = (m − 1)n i=1 2 4 J. Llibre & V. Pilyugina is denoted by πn (m). This number was introduced by Arnold [3] (also see Khovanskii [18]), and for low values of n is π2 (m) = m,   (1 + 3m2 )/4, if m is odd, π3 (m) =  0, if m is even, π4 (m) = (m + 2m3 )/3,   (27 + 50m2 + 115m4 )/192 if m is odd, π5 (m) =  0, if m is even, π6 (m) = (4m + 5m3 + 11m5 )/20,   (1125 + 1813m2 + 2695m4 + 5887m6 )/11520, if m is odd, π7 (m) =  0, if m is even, π8 (m) = (45m + 49m3 + 70m5 + 151m7 )/315 (for more detalls about the number πn (m), see the appendix of the paper of Cima, Gasull and Torregrosa [6]). For real homogeneous polynomial maps of degree m > 1, we have the following result. Theorem 1.2. The following statements hold. (a) Let f : Rn → Rn be a homogeneous polynomial map of degree m such that f (x) 6= 0, if x 6= 0. (a.1) Then, Per(f )= Per(F ) ∪ {1}, and (a.2) the topological degree of F is an integer d congruent with m module 2 satisfying |d| ≤ πn (m). (b) Let d be an integer congruent with m module 2 satisfying |d| ≤ πn (m). Then, there exists a homogeneous polynomial map of degree m such that f (x) 6= 0, if x 6= 0, for which the topological degree of Ff is d. In fact statements (a.2) and (b) of Theorem 1.2 follow directly from [3] and [18] respectively (for more details, see Corollary 1 of [18] and its proof). Here, we only will prove statement (a) of Theorem 1.2 in Section 3. The set of periods of the homogeneous polynomial maps f : R2 → R2 is charac- terized in the next theorem, but we need some definitions first. We recall the Sharkovskii ordering on the set N this total ordering from the largest one to the smallest one is 3, 5, 7, . . . , 2 · 3, 2 · 5, 2 · 7, . . . , 22 · 3, 22 · 5, 22 · 7, . . . , 2` · 3, 2` · 5, 2` · 7, . . . , 23 , 22 , 2, 1. For k ∈ N, the set S(k) is formed by k and all the elements of N, which follow k in the Sharkovskii ordering. Thus, for instance, S(3) = N. We also define the set S(2∞ ) = {1, 2, 4, 8, . . .}. Periods of Polynomial Homogeneous Maps 5 Let a, b ∈ R. For a ≤ b, we define the set M (a, b) = {k ∈ N : a < `/k < b}, i.e., M (a, b) is formed by the positive denominators of all rational numbers, which are in the interior of the interval [a, b]. We note that we do not assume that ` and k are coprime. Of course, M (a, b) = ∅, if a = b. Every continuous map F : S1 → S1 of degree 1 has associated a closed rotation interval [a, b], and eventually a and b can be equal (see [2] for a precise definition of the rotation interval). If a = `/k is rational with ` and k coprime, then we define the set S(a, r) = kS(r) for some r ∈ N ∪ {2∞ }. Here, kS(r) is the set formed by all the elements of the set S(r) multiplied by k. If a is irrational, then we define the set S(a, r) = ∅. Theorem 1.3. Let f : R2 → R2 be a homogeneous polynomial map of degree m > 1 such that f (x) 6= 0, if x 6= 0, and let d ∈ [−m, m] the topological degree of F = Ff . If d = 1, we denote by [a, b] the rotation interval of f . Then, Per(f ) = Per(F ) ∪ {1} and Per(F ) is equal to N, if d 6= −2, −1, 0, 1; N \ {2} or N, if d = −2; S(k) for some k ∈ N ∪ {2∞ }, if d = −1, 0; S(a, r) ∪ M (a, b) ∪ S(b, s) for some r, s ∈ N ∪ {2∞ }, if d = 1. Theorem 1.3 is proved in Section 4. From Theorem 1.3, it follows that Per(F ) 6= ∅, except if d = 1 and a = b ∈ / Q. In general, we can say almost nothing about the set of periods of the homo- geneous polynomial maps f : Rn → Rn , when n > 2, except if these maps are transversal. Then, we can say many things. Clearly, the map F = Ff , associated to a homogeneous polynomial map f : Rn → Rn with f (x) 6= 0, if x 6= 0, is of class C 1 . A C 1 map F : Sn−1 → Sn−1 is called transversal, if for all k ∈ N the graph of F k intersects transversally the diagonal of Sn−1 × Sn−1 at each point (x, x) such that x is a fixed point of f k . In other words, if the Jacobian matrix DF k (x) has not the eigenvalue 1 for all fixed point x of f k . Theorem 1.4. For n > 2, let f : Rn → Rn be a homogeneous polynomial map of degree m > 1 such that f (x) 6= 0, if x 6= 0, and let d ∈ [−πn (m), πn (m)] the topological degree of F = Ff . Assume that F is transversal. Then, Per(f ) = Per(F ) ∪ {1} and 1 ∈ Per(F ), if d = 0; Per(F ) can be empty, if n is odd and d = 1; 1 ∈ Per(F ), if n is even and d = 1; 1 ∈ Per(F ), if n is odd and d = −1; {1, 2} ∩ Per(f ) 6= ∅, if n is even and d = −1; Per(F ) ⊃ {1, 3, 5, 7, . . .}, and for every k, even if k 6∈ Per(F ), then Per(F ) ⊃ {k/2, 2k}, if d 6∈ {−1, 0, 1}. 6 J. Llibre & V. Pilyugina Theorem 1.4 is proved in Section 5. Here, we have considered homogeneous polynomial maps. However, if we con- sider polynomial maps, many other questions on these maps can be studied (see for instance [15, 16, 20, 28]). 2. Proof of Theorem 1.1 The set of invariant straight lines through the origin of a homogeneous polynomial map f : Cn → Cn of degree m has been studied by the authors in [23] and by Feng Rong [25] (also see [11]. The following result was proved by the authors under generic assumptions (see Theorem 1 of [23]), and in general in Theorem 1 of [25]. It also follows from Corollary 3.2 of [11]. Theorem 2.1. Let f : Cn → Cn be a homogeneous polynomial map of degree m. Assume that f has finitely many straight lines Lx such that f (Lx ) ⊂ Lx . Then, f has 1 + m + m2 + · · · + mn−1 = (mn − 1)/(m − 1) of these straight lines through the origin taking into account their multiplicities. From Theorem 2.1, it follows immediately the next result. Corollary 2.1. Let f : Cn → Cn be a homogeneous polynomial map of degree m such that f (x) 6= 0, if x 6= 0. Assume that f has finitely many invariant straight lines Lx such that f (Lx ) = Lx . Then, f has (mn − 1)/(m − 1) invariant straight lines through the origin taking into account their multiplicities. In 1964, Baker [4] completely described the periodic set of any polynomial map P : C → C. More precisely he proved: Theorem 2.2. Let P be a polynomial of degree at least two and suppose that P : C → C has no periodic points of period k. Then, k = 2, and P is conjugate to the polynomial z 2 − z. The proof of this theorem can also be found in [5] and [10]. In this last paper, Theorem 2.2 is proved using Lefschetz numbers. Proof. [Proof of Theorem 1.1] Statement (a) follows immediately from Corollary 2.1. For a map h having finitely many periodic points of every period we define F(k) as the number of fixed points of h` , and P(`) as the number of periodic points of period ` of h. Then, it is known that X P(`) = µ(r)F(`/r) (2.1) r|` (for more details, see [24]). Then, for the map f acting on the straight lines through the origin, we have that F(`) = (m`n − 1)/(m` − 1). In short, from (2.1), it follows immediately the statement (b) of Theorem 1.1. We shall prove the rest of the statements of Theorem 1.1 for k = 1, the proof for k > 1 is completely similar recalling that the degree of f k for k > 1 is mk . If f has infinitely many invariant straight lines, let Lx be one of them. Otherwise, since f (x) 6= 0, if x 6= 0, by Corollary 2.1 f has invariant straight lines. Let Lx be one invariant straight line of f . Periods of Polynomial Homogeneous Maps 7 Changing the coordinates from x = (x1 , . . . , xn ) ∈ Cn to y = (y1 , . . . , yn ) ∈ Cn doing a rotation through the origin of coordinates which pass the invariant straight line Lx to the y1 –axis of Cn , the homogeneous polynomial map f of degree m becomes a homogeneous polynomial map g of degree m having the y1 –axis invariant. Hence, g(y1 , 0, . . . , 0) = (ay1m , 0, . . . , 0) with a ∈ C \ {0}. In other words, the map f |Lx can be identified with the polynomial map P : C → C defined by P (z) = az m . Therefore, all the fixed points of P are the origin and all the roots of the equation z m−1 = (1/a)1/(m−1) . By Theorem 2.2, and since m > 1 and the polynomial map P (z) = az m is not conjugated to the map z 7→ z 2 − z, it follows that Per(P ) = N. Consequently, Per(f |Lx ) = N, and statement (c) of Theorem 1.1 is proved. Since f ` |Lx can be identified with the polynomial map P ` : C → C of the form ` ` P (z) = a(m −1)/(m−1) z m , all the periodic points of period ` or of period a divisor ` ` ` of ` of P are solutions of the equation a(m −1)/(m−1) z m = z. Then, for the map P the origin z = 0 is a fixed point, and the other periodic points with period a divisor of ` are the roots of the equation ` −1 1 zm = m` −1 . a m−1 Therefore, all the periodic points of f ` |Lx are on the circle centered at the origin of radius (1/a)1/(m−1) contained in Lx , the unique exception is the fixed point localized at the origin. Assume that f (Lx ) = Lx . Therefore, if y ∈ Lx \{0}, we have that f (y) = λ(y)y with λ(y) 6= 0, and consequently 2 +···+mk−1 f k (y) = λ(y)1+m+m y. Therefore, the fixed point 0 at the origin of the invariant straight line Lx = C is stable for the map f |Lx , and its bassin of attraction is the set of points y ∈ Lx such that |λ(y)| < 1. Note that for the points y in the circle of radius (1/a)1/(m−1) centered at the origin we have that their |λ(y)| = 1, because the set of all periodic points of f |Lx is dense in this circle. Hence, all the points of Lx in the interior of the region limited by the circle containing the periodic points of f |Lx tend to the origin under the iteration by f |Lx , and the points outside the region limited by this circle tend to infinity because for them |λ(y)| > 1. Therefore, all periodic points of the circle are repelling, and since they are dense in the circle, its closure (the Julia set) is the circle. In short, statement (d) is proved. Applying the formula (2.1) to the map f |Lx , we have that F(`) = m` . Therefore, it follows immediately the statement (e) of Theorem 1.1. Remark 2.1. If f (x) = λ(x)x for some λ(x) ∈ C \ {0}, then for every y ∈ C, we have that f (y) = λ(y)y, where λ(y) = cm−1 λ(x) with c = y/x. 3. Proof of Theorem 1.2 We denote by R+ the set of all positive real numbers. For each x ∈ Rn \ {0}, we define the ray Lx as Lx = {λx : for all λ ∈ R+ }. Therefore, it is clear that f (Lx ) = Lf (x) . 8 J. Llibre & V. Pilyugina Lemma 3.1. Let f : Rn → Rn be a homogeneous polynomial map of degree m > 1. If Lx = Lf (x) , then there is a unique fixed point in the ray Lx . Proof. Since the ray Lx is invariant by the map f , working as in the proof of Theorem 1.1, we can identify the map f |Lx to the polynomial map P : R+ → R+ defined by P (z) = az m , but now with a > 0. Since the equation az m = z has a unique real solution, namely z = (1/a)1/(m−1) , it follows that the map f |Lx has a unique fixed point. Hence, the lemma is proved. Under the assumptions of Lemma 3.1, it is easy to check that the fixed point y on the ray Lx = R+ = (0, +∞) is unstable. More precisely, the points smaller than y tends to the origin, and the points larger than y tends to +∞. Proof. [Proof of statement (a) of Theorem 1.2] We claim that if x ∈ Rn \ {0} is a k–periodic point of f , then x/||x|| ∈ Sn−1 is a k–periodic point of F . Now, we shall prove the claim. First, we shall see by induction that f j (x)   x Fj = for all j = 1, 2, . . . and x 6= 0. (3.1) ||x|| ||f j (x)|| For j = 1, the equality (3.1) follows from   x 1  x  f ||x|| m f (x) f (x) F =   = ||x|| 1 = . ||x|| ||f x || || ||x||m f (x)|| ||f (x)|| ||x|| Assume that (3.1) holds for 1, 2, . . . , j, and we shall prove it for j + 1. Indeed, we have  j  f (x)  x   j f (x)  f j ||f (x)|| F j+1 =F =  j  ||x|| ||f j (x)|| ||f f j (x) || ||f (x)|| 1 j+1 ||f j (x)||m f (x) f j+1 (x) = 1 = . ||f j (x)||m ||f j+1 (x)|| ||f j+1 (x)|| Hence, (3.1) is proved. Now, we prove the claim. Let x ∈ Rn \ {0} be a k–periodic point of f . First, from (3.1), we have that f k (x)   x x Fk = = . ||x|| ||f k (x)|| ||x|| Therefore, x/||x|| is fixed by F k . It remains to show that   x x Fj 6 = for all j = 1, . . . , k − 1. ||x|| ||x|| Assume that   x x Fj = for some j ∈ {1, . . . , k − 1}. ||x|| ||x|| Then, f j (x) and x are in the ray Lx , and this ray is invariant by f k , i.e., Lx = Lf k (x) . Therefore, f k has two fixed points in the ray Lx , namely f j (x) and x, which are in contradiction with Lemma 3.1. Periods of Polynomial Homogeneous Maps 9 In short, we have proved that if x ∈ Per(f ), then x ∈ Per(F ). On the other hand, let O be the origin of Rn , since f (O) = O, it follows that 1 ∈ Per(f ). Therefore, we have proved that Per(f ) ⊂ Per(F ) ∪ {1}. Now, we claim that if x ∈ Sn−1 is a k–periodic point of F , then there exists y ∈ Lx such that y is a k–periodic point of f . We prove the claim. From (3.1), we have f k (x) x = F k (x) = . ||f k (x)|| Hence, f k (x) = ||f k (x)||x. Then, Lx = Lf k (x) . Therefore, by Lemma 3.1, there exists y ∈ Lx such that f k (y) = y. From (3.1), if follows that Lf j (x) = LF j (x) for j = 1, 2, . . ., then since Lx = Ly the rays Lf j (y) = Lf j (x) = LF j (x) for j = 1, 2, . . . Since the rays LF j (x) are different for j = 0, 1, . . . , k − 1, it follows that y is a k–periodic point of f . Hence, the claim is proved. Consequently, Per(F ) ⊂ Per(f ). This completes the proof. 4. Continuous maps of the circle S1 The following result is proved in [2]. Theorem 4.1. Let F : S1 → S1 be a continuous map of degree d. If d = 1, we denote by [a, b], the rotation interval of F . Then, Per(F ) is equal to N, if d 6= −2, −1, 0, 1; N \ {2} or N, if d = −2; S(k) for some k ∈ N ∪ {2∞ }, if d = 0, 1; S(a, r) ∪ M (a, b) ∪ S(b, s) for some r, s ∈ N ∪ {2∞ }, if d = 1. Theorem 1.3 follows immediately from Theorem 4.1. 5. Transversal maps on the sphere Sn−1 For a C 1 transversal map f : Sn−1 → Sn−1 , we define its minimal set of periods in the class of C 1 transversal self–maps of Sn−1 as the set \ MPer(f ) := Per(g), g where g runs over all C 1 transversal self–maps of Sn−1 of the same degree than f . For C 1 transversal self–maps of Sn−1 , we have the following result (see Theorem 3 of [22]). Theorem 5.1. Let F : Sn−1 → Sn−1 be a transversal map of degree d. 10 J. Llibre & V. Pilyugina (a) If d = 0, then MPer(F ) = {1}. (b) If n − 1 is odd and d = 1, then MPer(F ) = ∅. (c) If n − 1 is even and d = 1, then MPer(F ) = {1}. (d) If n − 1 is odd and d = −1, then MPer(F ) = {1}. (e) If n − 1 is even and d = −1, then {1, 2} ∩ MPer(F ) 6= ∅. (f ) If d 6∈ {−1, 0, 1}, then MPer(F ) ⊃ {1, 3, 5, 7, . . .}, and for k even if k 6∈ MPer(F ), then MPer(F ) ⊃ {k/2, 2k}. Now, Theorem 1.4 is an easy corollary of Theorem 5.1. References [1] T. Aliashvili, Geometry and topology of proper polynomial mappings. 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