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0 1 0 ... 0  0 0 1 ... 0   . . .   . .. ..     0 ... 0 1     a0 ... an 1 − − −    of f (x) when f (x) is monic (i.e., an = 1). We prove that RMR( f (x)) is the union of the conjugacy classes of the roots of f (x), just like the roots of the companion n polynomial C( f (x)) = f (x) f (x) of f (x) ( f (x) = an x + +a ). The description of · ··· 0 LMR( f (x)) is more complicated, but a good description is provided when A = O. Dynamics of iterations of polynomials and rational functions over fields is a well-studied area of research (see [9, 11, 12] for instance). Some recent attempts have been made to generalize certain aspects of this theory to polynomials over the quaternions, such as [2] and [10]. More recently, the authors of this article attempted to generalize some aspects of this theory to general division rings and octonion algebras (see [5]). The study of octonion polynomials was also suggested in [7] in the context of polynomials over Lie algebras. It is important to note that the composition of such polynomials is non-associative, and in general f f (λ) , ◦ f ( f (λ)) for λ H, see Section 2 and [5] for details. ∈ In the second part of the paper, we show that if f (x) is a quadratic monic poly- nomial over an octonion algebra and f (α) = α, then f n(α) = α for all n N (this ◦ ∈ was shown to be true for all polynomials over associative division algebras but false over octonion algebras in [5]), and we determine when a given fixed point of such a polynomial over O or H is attracting, repelling or neutral. The last part provides more information about pseudo-periodic points, generalizing certain aspects from the theory of fixed points.

2 2. The Algebra of Octonion Polynomials

A Q over a field F is a central simple F-algebra of degree 2 (or equivalently, dimension 4). When char(F) , 2, it is generated over F by i and j subject to the relations i2 = α, j2 = β and i j = ji for some α, β F , and when − ∈ × char(F) = 2, it is generated by i and j subject to the relations i2 + i = α, j2 = β and ji j 1 = i+1 for some α F and β F . As a 4-dimensional real vector space, it is − ∈ ∈ × spanned by 1, i, j, i j, and i j is usually denoted by k. This algebra is endowed with a canonical involution, mapping each z = a + bi + c j + dk to z = a + b ji j 1 c j dk. − − − In the special case of F = R and α = β = 1 one obtains Hamilton’s algebra − of real quaternions, denoted by H. The canonical involution gives rise to a linear map (“the trace map”) Tr : Q F and a quadratic multiplicative map (“the norm → map”) Norm : Q F given by Tr(z) = z + z and Norm(z) = z z, respectively. → · When Q = H, since Tr(z) is twice the projection of z H on the line spanned by ∈ 1, in many sources one denotes “the real part” of z by (z), which stands for 1 Tr(z). ℜ 2 Furthermore, since Norm(z) is the square of the norm of z in the Euclidean metric, one usually denotes “the absolute value” of z by z , which stands for √Norm(z). | | We write (z) for z (z). ℑ −ℜ An octonion algebra over a field F is an algebra of the form A = Q Qℓ with ⊕ multiplication defined by

(q + rℓ)(s + tℓ) = qs + γtr + (tq + rs)ℓ, q, r, s, t Q, ∀ ∈ where Q is a quaternion algebra over F and γ F . The involution extends from ∈ × Q to this algebra by q + rℓ = q rℓ, and the trace and norm maps are defined in − the same manner. The norm form is still multiplicative, for this is a . Every element λ A satisfies λ2 Tr(λ)λ + Norm(λ) = 0. The algebra ∈ − A is a division algebra (i.e., every nonzero element has an inverse) if and only if the norm form is anisotropic. The algebra A is an , i.e., every two elements generate an associative subalgebra, but A itself is nonassociative. The algebra satisfies the Moufang laws (zx)(yz) = z(xy)z, z(x(zy)) = (zxz)y and ((xz)y)z = x(zyz). For more information about these algebras see [13] and [8]. We write [x, y] for xy yx. − By taking Q = H and γ = 1 one obtains the real octonion algebra O, and the − notions of absolute value and real part extend to this algebra as well. The algebra of polynomials over an octonion algebra A over a field F is de- fined to be A[x] = F[x] F A. Therefore, the indeterminate x commutes with all ⊗ the elements of F, and is in the center of A[x]. We define the composition for poly- n n nomials f (x) = an x + + a1 x and g(x) by f g(x) = an(g(x)) + + a1g(x) + a0. n ··· n ◦ (n 1) 1··· We define f ◦ (x) recursively by f ◦ (x) = f f ◦ − (x) and f ◦ (x) = f (x). We n n (n ◦1) 1 define f ∗ (α) recursively by f ∗ (α) = f ( f ∗ − (α)) and f ∗ (α) = f (α), where the

3 n substitution of an element α A in f (x) = an x + + a1 x + a0 is defined by n ∈ ··· f (α) = anα + + a α + a . Note that the substitution map f (x) f (α) is not ··· 1 0 7→ an algebra homomorphism from A[x] to A. In particular, unlike the commutative n n case, there is in general no equality between f ◦ (α) and f ∗ (α) (see [5]). We say that α is a “fixed point” of f (x) if f n(α) = α for every n N. It ◦ ∈ was proven in [5] that if f (α) = α where f is defined over a division algebra, then α is a fixed point. As we demonstrate in Section 4, this is also true for monic quadratic octonion polynomials. This is in contrast to the case of general octonion polynomials, for which this statement is false, as shown in [5]. In complex dynamics (see [9, §4], or [1, §6.1]), we define a fixed point to be “attracting” if f (α) < 1, “repelling” if f (α) > 1, “neutral” if f (α) = 1 and | ′ | | ′ | | ′ | “super-attracting” if f ′(α) = 0. The derivative for polynomials over H or O is not 2 quite well-defined, and if one considers the “formal derivative” (i.e., (x +Ax+B)′ = 2x + A), then this does not characterize well the dynamical properties we expect from attracting, repelling or neutral fixed points, as we shall see. Instead, we will define the classification of fixed points by the expected dynamical properties they should have (as in the classical case). In this article we call a fixed point α of f (x) O[x] an “attracting” fixed point if there exists a neighborhood S of α (a ∈ neighborhood should contain an open ball centered at α) and a positive c smaller than 1 such that for every λ S , f (λ) α < c λ α . In particular, ∈ | − | · | − | the sequence f n(λ) converges to α. We say that α is “repelling” if there { ∗ }n∞=1 exists a neighborhood S of α and a positive real number c greater than 1 such that for every λ S , f (λ) α > c λ α . We call α “ambivalent” if it is ∈ | − | · | − | neither attracting nor repelling (the term “neutral” does not carry well to dynamics of quaternion polynomials, as a neighborhood of an ambivalent fixed point can contain points whose trajectories are attracted to the fixed point as well as points whose trajectories are repelled by the fixed point). The question of classifying the fixed points of some quadratic polynomials over H was addressed in [2] and [10] with a special emphasis on f (x) = x2+C for C H. ∈ This family of quadratic polynomials is natural in complex dynamics, as one can easily show that any monic quadratic polynomial p(x) = x2 + Bx +C with B, C C ∈ is equivalent to a polynomial of the form f (x) = x2 +C with C C under the ring ′ ′ ∈ automorphism x x B . This map, however, is not a ring automorphism for H[x] 7→ − 2 when B is not central, therefore it is not enough to consider only quadratic monic 2 polynomials of the form f (x) = x + C′ over Quaternion or Octonion algebras.

3. Roots of scalar multiples of octonion polynomials

Set a field F and an octonion division algebra A over F, and let f (x) be an element of A[x]. Following [4], the roots of f (x) can be obtained from the roots of

4 its companion polynomial C( f (x)) = f (x) f (x), whose coefficients lie in F. The · roots of C( f (x)) are the conjugacy classes of the roots of f (x). Each conjugacy class is characterized by the trace T and norm N of its members. By the identity z2 Tr(z)z + Norm(z) = 0, the equality f (λ) = 0 reduces to a linear equation − E(N, T)λ + G(N, T) = 0 when we know that Tr(λ) = T and Norm(λ) = N. Then there are two options - either E(N, T) = 0, in which case G(N, T) must be zero as well, and then the entire conjugacy class is a subset of R( f (x)), or E(N, T) , 0, in which case the only element from the conjugacy class that is in R( f (x)) is E(N, T) 1G(N, T). − − Remark 3.1. The companion polynomial of g(x) = f (x)c for c A is C( f (x)c) = ∈ × f (x)c f (x)c = C( f (x)) Norm(c), and thus has the same roots as C( f (x)). The same · · holds true for the companion polynomial of h(x) = c f (x). Therefore RMR( f (x)) ⊆ R(C( f (x))) LMR( f (x)). For each conjugacy class of roots of C( f (x)) with norm ⊇ N and trace T, the reduction of g(λ) = 0 to a linear equation gives (E(N, T)c)λ + G(N, T)c = 0, and thus when E(N, T) , 0, (c 1E(N, T) 1)(G(N, T)c) is the − − − root of g(x) in this conjugacy class. Similarly, the reduction of h(λ) = 0 to a linear equation gives (cE(N, T))λ + cG(N, T) = 0, and thus when E(N, T) , 0, (E(N, T) 1c 1)(cG(N, T)) is the root of h(x) in this conjugacy class. − − − Lemma 3.2 ([4, Remark 5.3]). Given an octonion algebra A over a field F, if µ, λ A are conjugates, then there exists some δ A× of Tr(δ) = 0 for which ∈ 1 ∈ µ = δλδ− . Theorem 3.3. For any f (x) A[x],RMR( f (x)) = R(C( f (x))). ∈ Proof. The inclusion RMR( f (x)) R(C( f (x))) is immediate from Remark 3.1. ⊆ It is enough to show the opposite inclusion, and for that one needs to show that for any root λ of f (x), all its conjugates are in RMR( f (x)). Let λ be a root of 1 f (x) of norm N and trace T, and let µ be a conjugate of λ. Then µ = δλδ− for some δ A of Tr(λ) = 0. If E(N, T) = 0, then all the conjugates of λ, ∈ × including µ are roots of both f (x)γ for any γ A , and so µ RMR( f (x)). Suppose ∈ × ∈ E(N, T) , 0. Then λ = E(N, T) 1G(N, T). Take g(x) to be f (x)δ 1. Then − − − (δE(N, T) 1)(G(N, T)δ 1) is a root of g(x). But δ 1 is a (central) scalar multiple − − − − of δ, because Tr(δ) = 0, and so it follows from the Moufang identity (zx)(yz) = z(xy)z that (δE(N, T) 1)(G(N, T)δ 1) = δ( E(N, T) 1G(N, T))δ 1 = δλδ 1 = µ. − − − − − − − Therefore, µ RMR( f (x)).  ∈ Theorem 3.4. Let λ be a root of f (x) A[x]. Write T and N for its trace and norm, ∈ and set E = E(N, T) and G = G(N, T). Then, when E = 0, the entire conjugacy class of λ is contained in LMR( f (x)), and when E , 0 and E and G live in a

5 quaternion subalgebra Q of A, the algebra A decomposes as a Cayley doubling A = Q Qℓ with ℓ2 = γ F , and the intersection of LMR( f (x)) with [λ] is ⊕ ∈ × 1 Norm(a)E 1G γNorm(b)GE 1 + (b[G, E 1]a)ℓ) : a, b Q . Norm(a + bℓ) − − − ( ·  −  ∈ ) Proof. The fact that [λ] LMR( f (x)) when E(N, T) = 0 follows from Remark ⊆ 3.1. Suppose E(N, T) , 0 and E(N, T) and G(N, T) live in a quaternion subalgebra Q of A. Then A is a Cayley doubling Q Qℓ of Q, with ℓ2 = γ for some γ F . ⊕ ∈ × Write c = a+bℓ where a, b Q. Then c 1 = 1 (a bℓ), and so (E 1c 1)(cG) = ∈ − Norm(c) − − − 1 (E 1a (bE 1)ℓ)(aG + (bG)ℓ) = 1 (Norm(a)E 1G γNorm(b)GE 1 + Norm(c) − − − Norm(c) − − − (b(GE 1 E 1G)a)ℓ), and the statement follows.  − − − Remark 3.5. When char(F) , 2, the condition that E and G always live in a quaternion subalgebra Q of A is always satisfied. Moreover, when A = O and E , 0, Q is an isomorphic copy of H, and LMR( f (x)) [λ] simplifies as ∩ 1 1 1 xE− G + (1 x)GE− + zℓ : 0 x 1, z Q, Norm(z) = x(1 x) Norm([G, E− ]) . n − ≤ ≤ ∈ − · o Example 3.6. The polynomial f (x) = x2 + ix i j + 1 has companion polynomial − C( f (x)) = x4 +3x2 +2 whose complex roots are i and √2i. The conjugacy class of i in O is characterized by trace 0 and norm 1. When one reduces the equation f (λ) = 0 to a linear equation by the rule λ2 = 1, one obtains the linear equation iλ i j = 0. − − Therefore E = i, G = i j, Q = H and λ = j. The intersection of LMR( f (x)) with − [ j] is therefore (1 2x) j + zℓ : 0 x 1, z H, Norm(z) = 4x(1 x) . { − ≤ ≤ ∈ − }

4. Fixed Points of Octonion Polynomials

Here we study the behaviour of fixed points of polynomials in A[x] under the condition that the fixed point and the coefficients of the polynomial belong to one associative sublagebra of the octonion algebra A. For a polynomial f (x) whose coefficients lie in an associative subalgebra of A, we write f t(x) for the product f (x) ... f (x). For α A we write f t(α) for the · · ∈ t times t t substitution of α into f (x), and| ( f (α{z)) for the} t-th power of f (α). In general the values of f t(α) and ( f (α))t are not the same.

Theorem 4.1 (cf. [5, Section 4]). Let A be an octonion algebra over F, f (x) = m n am x + + a x + a and g(x) = bn x + + b x + b in A[x] and α A, such that ··· 1 0 ··· 1 0 ∈ α, a0,..., am, b0,..., bn belong to an associative subalgebra of A. 1. If g(α) commutes with α, then h(α) = f (α) g(α) where h(x) = f (x)g(x). ·

6 2. If f (α) commutes with α, then f t(α) = ( f (α))t for any t N. ∈ 3. If α commutes with f n(α) for all n N, then f n(α) = f n(α) for all n N. ∗ ∈ ◦ ∗ ∈ 4. If f (α) = α then f n(α) = α for all n N. ◦ ∈ m r m n r+s Proof. First, h(x) = f (x) g(x) = r=0(ar x ) g(x) = r=0 s=0 arbs x . Hence, m n r+·s m n · s r m r m r h(α) = r=0 s=0 arbsα = r=0Par( s=0 bsα )α = Pr=0 aPr(g(α))α = r=0 arα g(α) = m i ( r=0 aPiα )g(Pα) = f (α) g(α).P The secondP statementP is proven by inductionP on t 1 · t 1 t.P Write g(x) = f − (x), h(x) = f (x)g(x), and assume g(λ) = ( f (λ)) − , which commutes with λ. Then by the first statement, h(λ) = f (λ) g(λ) = ( f (λ))t. · (n 1) The third statement is proven by induction on n. Write g(x) = f ◦ − (x). Then n m t f ◦ = f g(x) = t=0 atg (x). Since the coefficients of g(x) and f (x) and α belong ◦ t to an associativeP subalgebra of A, the coefficients of g (x) belong to that subalge- t bra too, and therefore substituting α in the polynomial atg (x) is equal to at times t (n 1) t g (α), giving at f ∗ − (α) by the induction hypothesis and the second statement. n  m  (n 1) t n Therefore, f ◦ (α) = t=0 at f ∗ − (α) = f ∗ (α). The fourth statement follows n N from the third, becauseP f (α) = α implies f ∗ (α) = α for all n , and therefore α ∈ commutes with f n(α) for all n N.  ∗ ∈ Corollary 4.2. If A is an octonion algebra over F, f (x) is a quadratic monic poly- nomial in A[x], and α A satisfies f (α) = α, then α is a fixed point of f (x). ∈ Proof. Write f (x) = x2 + Bx + C. Then f (α) = α2 + Bα + C = α, which means C = α α2 Bα. Since B and α generate an associative subalgebra of A, and C − − belongs to that subalgebra, the conditions of Theorem 4.1 are met, and therefore f n(α) = α for any n N.  ◦ ∈

5. Classification of Fixed Points

Let f (x) = x2 + Bx + C be a quadratic monic polynomial over H or O. By Corollary 4.2, a fixed point is a root of g(x) = f (x) x. Algorithms for finding − such roots were provided in [6] (over H) and [4] (over O).

Theorem 5.1. Let f (x) = x2 + Bx + C be a polynomial over H or O with a fixed point α, and write M = (2α + B)2 + ( (α + B) + (α) )2 and m = ℜ |ℑ | |ℑ | (2α + B)2 + ( (α + B) (αp) )2. ℜ |ℑ |−|ℑ | p 1. If M < 1, then α is attracting. 2. If m > 1, then α is repelling. 3. If m 1 and 1 M, then α is ambivalent. ≤ ≤

7 2 1 Proof. Write λ = α + β. Then f (λ) = α + (α + B)β + βα + β . Setα ˜ = βαβ− , and then f (λ) α = (α+ B+α˜)β+β2. Sinceα ˜ is in the conjugacy class of α, and in fact − for the right choice of β it can be any element in this conjugacy class, the absolute value of α + B + α˜ ranges between m and M. If M < 1, then clearly there exists a small enough neighborhood S of α and a positive real number c smaller than 1 such that for any λ = α + β S , we have f (λ) α < c β . If m > 1 then clearly ∈ | − | | | there is a small enough neighborhood S of α and a positive real number c greater than 1 such that for any λ = α + β S , we have f (λ) α > c β . If 1 M, then in ∈ | − | | | ≤ any neighborhood S of α, if one fixes a positive real number c smaller than 1, then by picking a small enough β for which the absolute value of α+ B+α˜ is exactly M, then f (λ) α will be greater than c β , and therefore α is not attracting. Similarly, | − | | | if m 1, then in any neighborhood S of α, if one fixes a positive real number c ≤ greater than 1, then by picking a small enough β for which the absolute value of α + B + α˜ is exactly m, then f (λ) α will be smaller than c β , and therefore α is | − | | | not repelling.  This means that there are certain complex polynomials with attracting fixed points that do not remain attracting when extending their domain to H or O. Example 5.2. Consider f (x) = x2 + ix 1 i 1 over H or O. Here B = i. It has − 2 − 4 α = i as a fixed point and f (α) = 2α + i = 0, so as a polynomial over C the fixed − 2 ′ point α is super-attracting (so in particular it is attracting). However, M = 1, so α is not an attracting fixed point when considering f (x) as a quaternionic polynomial. Here one can produce explicitly an element in every neighborhood of α whose orbit does not converge to α. Take λ = 1 i + r + γ where r C and γ C j 0 . − 2 ∈ ∈ \{ } Then the projection of f (λ) α on C j is (2 (r) + i)γ, which is of absolute value − ℜ greater or equal to γ . Therefore, the orbit of λ does not converge to α. | | 6. Pseudo-Periodic Points Given a polynomial f (x) over a division algebra or octonion algebra D, and λ D, we say that α is a pseudo-periodic point of f (x) if f n(α) = α for some ∈ ∗ n N. The minimal n for which this is satisfied is called the order of the periodic ∈ point. By [5], pseudo-periodic points need not be periodic. For D = O or H, a pseudo-periodic point α of f (x) of order n over D is called “attracting” if there exists a positive real constant c < 1 and a neighbourhood S of α such that every λ S satisfies f n(λ) α < c λ α . ∈ | ∗ − | | − | Theorem 6.1. Given a monic quadratic polynomial f (x) = x2 + Bx + C over O or H n 1 i with a pseudo-periodic point α of order n, if i=−0 Mi < 1, where αi = f ∗ (α) 2 2 and Mi = (2αi + B) + ( (αi + B) + (αi) ) Qfor all i 0, 1,..., n 1 , then α ℜ |ℑ | |ℑ | ∈{ − } is attracting.

8 Proof. Write λ = α + β. Then

f (λ) f (α) = (α+B+α˜)β+β2 β ( ( (2α + B)2 + ( (α + B) + (α) )2)+ β ), | − | | | ≤ | |· p ℜ |ℑ | |ℑ | | | f (λ) f (α) and so | − | ( (2α + B)2 + ( (α + B) + (α) )2) + β . Therefore, λ α ≤ ℜ |ℑ | |ℑ | | | | − | p n 1 + + n 1 n 1 f n(λ) α − f (i 1)(λ) f (i 1)(α) − f (λ ) f (α ) − ∗ = ∗ ∗ = i i + | − | | i − i | | − | ( Mi βi ), λ α f (λ) f (α) λi αi ≤ | | | − | Yi=0 | ∗ − ∗ | Yi=0 | − | Yi=0 p i i where we take λi = f (λ),αi = f (α) and βi = λi αi for all i 0, 1,..., n 1 . ∗ ∗ − ∈{ − } Notice that βi βi 1 ( √Mi + βi 1 ) for i 1,..., n 1 , so that by choosing a | | ≤ | − | | − | ∈ { − } small enough neighborhood, the values of β0 ,..., βn 1 can be made arbitrarily n 1 | | | − | n 1 small. Since i=−0 Mi < 1, for a small enough neighborhood S also i=−0 ( √Mi +  βi) < 1, and theQ statement follows. Q Remark 6.2. In [2] it was shown that for f (x) = x2 + C for C H, a pseudo- n 1 ∈ i periodic point α of order n is attracting when i=−0 2α < 1 where αi = f ∗ (α) for n 1 | | i 0,..., n 1 (the product i=−0 2α is calledQ the multiplier in classical complex ∈{ − } (n) dynamics, defined to be the valueQ of the derivative of f ◦ (x) at α). This is recov- 2 2 ered in Theorem 6.1 because when B = 0, we have (2αi) + ( (αi) + (αi) ) = ℜ |ℑ | |ℑ | 2αi . p | |

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