This is a preprint of: “Dynamics of the secand map near infinity”, Antoni Garijo, Xavier Jarque, J. Difference Equ. Appl., vol. 28(10), 1334–1347, 2023. DOI: [10.1080/10236198.2022.2044476] DYNAMICS OF THE SECANT MAP NEAR INFINITY ANTONIO GARIJO AND XAVIER JARQUE Abstract. We investigate the root finding algorithm given by the secant method applied to a real polynomial p as a discrete dynamical system defined on R2 . We extend the secant map to the real projective plane RP2 , proving the existence of an attracting fixed point on the line of infinity, which allow us to better understand the dynamics of the secant method near ∞. Moreover, this fixed point only depends on the degree of the polynomial p. Keywords: Root finding algorithms, iteration, secant method. 1. Introduction and preliminaries Root finding algorithms are not only an efficient way to find numerical solutions of non linear equations which cannot be solve explicitly but a fruitful source of interesting dynamical systems. The common idea behind any such algorithm is to construct sequences converging to the solutions of the equation, and these sequences correspond to orbits of discrete dynamical systems. Let X be a topological space. Roughly speaking a discrete dynamical system over X (known as phase space) is a map f ∶ X → X and the orbits induced by this map starting at x0 ∈ X, {xn ∶= f n (x0 )}n∈N . The main goal is to describe the phase portrait, that is the description of the asymptotic behaviour of those orbits when x0 runs over all X. We say that ζ in X is a fixed point if f (ζ) = ζ. We say that ζ is attracting or repelling depending if all nearby seeds correspond to orbits converging or divergingto ζ. It is well known that fixed (as well as periodic) points play a key role to understand the global dynamics, specially when f models a root finding algorithm. In the case that ζ in X is an attracting fixed point we define its basin of attraction by A (ζ) = {x ∈ X ∣ f n (x) → ζ, as n → ∞}, or, in other words, A(ζ) is the maximal set where orbits converge to ζ under iteration. The connected component of A (ζ) which contains x0 is called the immediate basin of attraction of ζ and it is denoted by A⋆ (ζ). Why discrete dynamical systems are related with root finding algorithms is almost direct. Suppose we want to solve the nonlinear equation p(x) = 0. A root finding algorithm can be understood as a discrete dynamical system fp ∶ X → X so that its orbits {xn ∶= fpn (x0 )}n∈N , converge to the solutions of p(x) = 0 for most initial conditions x0 ∈ X. For instance, studing the global phase portrait associated to fp gives valuable information on how to choose the initial conditions to find all solutions of p(x) = 0 (see [HSS01] for the case that p is a polynomial with complex coefficients). To simplify the exposition we focus on the case that the nonlinear equation p = 0 is given by a polynomial with complex coefficients. The most well-known and universal root finding Date: January 20, 2022. This work has been partially supported by MINECO-AEI grants PID2020-118281GB-C32 and PID2020- 118281GB-C33, and the AGAUR grant 2017SGR1374. 1 2 ANTONIO GARIJO AND XAVIER JARQUE algorithm to find out the roots of p is Newton’s method defined as the iterates of the Newton’s map N ∶= Np (z) = z − p(z) p′ (z) (1) . Observe that p(α) = 0 if and only if Np (α) = α. Moreover, easy computations show that ∣Np′ (α)∣ < 1 and so the roots of p are attracting fixed points of Np , in fact if α is a simple root then Np′ (α) = 0, and one of the major interest is to study the basins of attraction of the different roots of p as well as their boundaries which are contained in the Julia set of Np . A first step is to extend the Newton map to the point at infinity, that is, to extend the phase space from C to Ĉ = C ∪ {∞}, where Ĉ denotes the Riemann sphere. Using appropriate charts on Ĉ we can see that z = ∞ is always a repelling fixed point of Np . It is worth to be noticed that Newton’s method was the starting point of holomorphic dynamics when Cayley used dynamical systems to understand the phase portrait of Newton’s method applied to low degree polynomials (see [Cay79a, Cay79b, Cay80]). Because of the importance of this universal root finding algorithm there is a wide literature on different aspects of Newton’s method, including recent results for the non polynomial case. See for instance [HSS01, Shi09, BFJK14, BFJK18]). An alternative to Newton’s method is the Secant’s method. Although it admits a complex variable version (see [BF19] and Section 4 in this paper) in this work we focus on the (real) Secant’s method applied to real polynomials of the form (2) p(x) = ak xk + . . . + a1 x + a0 , ak ≠ 0, with k ≥ 3. We also assume that p has n real simple roots α0 < α1 < . . . < αn−1 with n ∈ {0, . . . , k}. The roots α0 and αn−1 are called external roots and the rest internal roots. The secant’s method is the root finding algorithm defined as the iterates of the Secant’s map S ∶= Sp ∶ R2 ↦ R2 given by S (x, y) y S∶( )↦( 1 ) = ( y − p(y) y−x ) = ( yp(x)−p(y)x ) . x y S2 (x, y) (3) y p(y)−p(x) p(x)−p(y) Observe that in contrast to Newton’s method, the phase space of the Secant’s method is two dimensional: R2 or C2 in its complex version. To visualize this, consider the Chebychev’s polynomial p4 (x) = 1 − 8x2 + 8x4 having all four single roots in the interval (−1, 1). So, given a root z = α of p4 Newton’s method Np (see Figure 1(a)) exhibits a fixed point located a z = α (in this case real) while the Secant’s method Sp (see Figure 1(b)) has a fixed point at (x, y) = (α, α) ∈ R2 . We show the two dynamical planes: the complex plane C for Newton’s method and the real plane R2 for the Secant’s method. We plot with the same colour points converging to the same root using both algorithms. However for Newton’s method the attracting fixed points are located on the real line {z = x + iy ∈ C ∣ y = 0} while for the Secant’s method the fixed points are located on the diagonal line {y = x} ⊂ R2 . Similarly Figures 1(c-d) illustrate the same phenomena for p5 (x) = 5x − 20x3 + 16x5 . We refer to [BF19, GJ19, GJ20], and references therein, for a general background of the Secant’s method applied to a polynomial p. In this paper we emphasize on the behaviour of the Secant’s method near infinity. As we already noticed, in the Newton’s case we might use the complex structure of S2 to show that via the local chart ϕ(z) = 1/z we have that the Newton’s map is conjugated near infinity to DYNAMICS OF THE SECANT MAP NEAR INFINITY 3 (a) Dynamical plane of the Newton map applied to (b) Dynamical plane of the Secant map applied to p4 (x) = 1 − 8x2 + 8x4 . p4 (x) = 1 − 8x2 + 8x4 . (c) Dynamical plane of the Newton map applied to (d) Dynamical plane of the Secant map applied to p5 (x) = 5x − 20x3 + 16x5 . p5 (x) = 5x − 20x3 + 16x5 . Figure 1. Dynamical planes of the Newton’s map (left) and the Secant’s map (right) map applied to the Chebychev polynomials p4 (x) = 1 − 8x2 + 8x4 (top) and p5 (x) = 5x − 20x3 + 16x5 (bottom). The range of all the pictures is [−1.5, 1.5] × [−1.5, 1.5]. the map Gp (z) = (ϕ ○ Np ○ ϕ−1 )(z) = 1 Np (1/z) 4 ANTONIO GARIJO AND XAVIER JARQUE near the origin. Thus, Newton’s method exhibits a repelling fixed point at z = ∞ with multiplier Np′ (∞) ∶= G′p (0) = k/(k − 1) > 1. Moreover it is proven in [HSS01] that z = ∞ belongs to the boundary of all immediate basins of attraction of all fixed points of Np , or equivalently, the immediate basins of attraction of all fixed points (roots of p) of Np are unbounded. This crucial property is used in [HSS01] to determine a universal set of initial conditions, only depending on k (the degree of p), from which we can find all roots of p at once. Our goal is to extend the Secant’s map to infinity and see the potential consequences and numerical applications. In this case, however, we cannot add just a point as we did with the Newton case. Instead we use the compactification of R2 given by the projective plane RP2 . A first step is to extend S, using homogeneous coordinates, to a map Ŝ on RP2 . Once this is done we observe that Ŝ itself induce a map Ŝ∞ over the line at infinity of RP2 . The expression of Ŝ∞ only depends on the degree k of the polynomial p. In Lemma 3.2 we study the dynamics of Ŝ∞ and we show that, qualitatively, it only depends on the parity of k. Alternatively, the map Ŝ∞ at a given point on the line at infinity can also be defined as the limit behaviour of S over a line ending at this point-direction. This is well defined at all point-directions except the horizontal one since (4) lim Sp (x, y0 ) = (y0 , y0 ), x→∞ and so Ŝ([1 ∶ 0 ∶ 0]) would be a point in R2 . In Figure 2 we show the dynamical plane of the Newton’s method and the Secant’s method applied to the Chebyshev polynomials p4 (x) = 1 − 8x2 + 8x4 and p5 (x) = 5x − 20x3 + 16x5 , but now we focus our attention on the behaviour of S near infinity (compare with Figure 1). To do so, we show the dynamical planes inside a large circle centerer at the origin. In Figures 2(a) and (c) we can see that the immediate basins of attraction of the roots of p approach infinity forming wide accesses or sectors and this was the basic idea behind the mentioned results in [HSS01] to determine a universal set of initial conditions. In contrast, the Secant’s method near infinity seems to be only related with the largest (red) and smallest (blue) roots of p with the exception of two (k odd) or three (k even) points. The reason for this to happen is a direct consequence Lemma 3.2 and it is explained in Section 5. Our main result is the study of the global dynamics of Ŝ∞ . That is, to show that S defines a map over the line at infinity of RP2 and that such dynamics only depends on k, the degree of the polynomial p. Let k ≥ 3 and let ϕk ∶ R → R be the real map defined by ζ k−1 − 1 ζ k−2 + ζ k−3 + . . . + ζ + 1 ϕk (ζ) = = k−1 ζk − 1 + ζ k−2 + . . . + ζ + 1 (5) . ζ Theorem A. Let p be a degree k polynomial and ∞ (6) ∆ = ⋃ ϕ−n k (−1). n=0 Then, the following statements hold. (a) If k is odd, then there exists a unique fixed point [1 ∶ ηk ∶ 0] in `∞ for Ŝ∞ , which is attracting. Moreover, Ŝ∞ n ([1 ∶ ζ ∶ 0]) → [1 ∶ ηk ∶ 0] as n → ∞, for all ζ ∈ R ∖ {0, −1}. (b) If k is even, then Ŝ∞ has exactly two fixed points: one attracting [1 ∶ ηk ∶ 0] and one repelling [1 ∶ τk ∶ 0]. Furthermore, we have that Ŝ∞ n ([1 ∶ ζ ∶ 0]) → [1 ∶ ηk ; 0] as n → ∞, for all ζ ≠ {0} ∪ {τk } ∪ ∆. DYNAMICS OF THE SECANT MAP NEAR INFINITY 5 The paper is organized as follows. In Section 2 we introduce homogeneous coordinates to define Ŝ in RP2 . In section 3 we prove Theorem A. In section 4 we extend Ŝ in CP2 and study the global dynamics of this extension at infinity. In Section 5 we conclude some numerical considerations. We would like to thank the anonymous referees of a previous version of this paper. We learn from their reports a better way to approach our results. 2. The secant map on RP2 In this section we extend the Secant map S to the infinity by considering its expression over the whole the projective plane RP2 = R2 ⊔ RP1 , where ⊔ denotes a disjoint union. In this way the line at infinity is represented by RP1 or `∞ . In the complex version we use the notation CP2 = C2 ⊔ CP1 instead of RP2 and in this case the line at infinity is the Riemann sphere. Recall that RP2 is a compact, non orientable space which can be associated to the set of lines through the origin in R3 . On the one hand we identify every line through the origin not contained in the plane z = 0 with the intersection point between this line and the plane z = 1. So, we might parametrize these points by [x ∶ y ∶ 1] (this is precisely R2 in the definition of RP2 above). See Figure 3(a). The set of lines excluded in this construction, that is the lines contained in z = 0, form the so called line at infinity of the real projective plane. The coordinate [0 ∶ ζ ∶ 0] of a point in `∞ is determine by the slope of the line ζ ∶= y/x when x ≠ 0 and the remaining point directly by [0 ∶ 1 ∶ 0]. See Figure 3(b). Notice that antipodal points are identified. All together determine the coordinate of each point in RP2 , also called homogeneous cordinates ⎧ ⎪ [x ∶ y ∶ 1] with (x, y) ∈ R2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ RP = R ⊔ RP = ⎨ [1 ∶ ζ ∶ 0] with ζ ∈ R 2 2 ⎪ (7) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [0 ∶ 1 ∶ 0] If [x ∶ y ∶ z] are the homogenous coordinates in RP2 , the expression of the Secant map in (3) writes, in the z = 1 coordinate chart (that is, (x, y) ↔ [x ∶ y ∶ 1]), as a rational map Ŝ given by y/z p (x/z) − x/z p (y/z) Ŝ[x ∶ y ∶ z] = Ŝ[x/z ∶ y/z ∶ 1] = [y/z ∶ ∶ 1] (8) p (x/z) − p (y/z) = [y (p (x/z) − p (y/z)) ∶ y p (x/z) − x p (y/z) ∶ z (p (x/z) − p (y/z))] . So, denoting by q(t, z) = z k p(t/z) = ak tk + ak−1 tk−1 z + . . . + a0 z k direct computations from (8) give the following expression of Ŝ as a 3-tuple of homogeneous polynomials Ŝ[x ∶ y ∶ z] = [y (p (x/z) − p (y/z)) ∶ y p (x/z) − x p (y/z) ∶ z (p (x/z) − p (y/z))] = z k [y (p (x/z) − p (y/z)) ∶ y p (x/z) − x p (y/z) ∶ z (p (x/z) − p (y/z))] = [y (q (x, z) − q (y, z)) ∶ y q (x, z) − x q (y, z) ∶ z (q (x, z) − q (y, z))] (9) = [y(xk − y k ) + O(z) ∶ (yxk − xy k ) + O(z) ∶ O(z)] 6 ANTONIO GARIJO AND XAVIER JARQUE τ4 −1 ● ● η4 ● (a) Dynamical plane of the Newton map applied to (b) Dynamical plane of the Secant map applied to T4 (x) = 1 − 8x2 + 8x4 . T4 (x) = 1 − 8x2 + 8x4 . −1 ● η5 ● (c) Dynamical plane of the Newton map applied to (d) Dynamical plane of the Secant map applied to T5 (x) = 5x − 20x3 + 16x5 . T5 (x) = 5x − 20x3 + 16x5 . Figure 2. Dynamical planes of the Newton (left) and Secant (right) map applied to the Chebychev polynomials T4 (x) = 1 − 8x2 + 8x4 (top) and T5 (x) = 5x − 20x3 + 16x5 (bottom). We show the dynamical plane inside the disk centered at the origin with radius 10, points outside this disk are colored in black. Remark 1. The indeterminacy points of Ŝ are given by 3-tuples [x ∶ y ∶ z] such that Ŝ[x ∶ y ∶ z] = [0 ∶ 0 ∶ 0]. According to (9) we have DYNAMICS OF THE SECANT MAP NEAR INFINITY 7 y z [1 ∶ m2 ∶ 0] [1 ∶ m1 ∶ 0] z=1 1 [1 ∶ 0 ∶ 0] x y z=0 [1 ∶ m3 ∶ 0] x [0 ∶ 1 ∶ 0] (a) The space of all lines through the origin in (b) The space of all lines through the origin R3 not contained in the plane z = 0. in the plane z = 0. Figure 3. A model for the real projective plane RP2 = R2 ⊔ `∞ (a). The line of infinity `∞ (b). (a) Finite indeterminacy points (z = 1 in homogeneous coordinates). They correspond to points (x, y) such that x ≠ y and p(x) = p(y) = 0. For a discussion of finite indeterminacy points we refer to [GJ19] where they are called focal points. (b) Points at infinity (z = 0 in homogeneous coordinates) such that y = 0 and x ≠ 0, or x = y and xy ≠ 0 (that is, {[1 ∶ 0 ∶ 0], [1 ∶ 1 ∶ 0]}, see the last expression in (9)). 3. Proof of Theorem A We split the proof of Theorem A in two technical lemmas. First we observe that Ŝ defines a map over the line at infinity. Lemma 3.1. Consider a polynomial p as defined in (2). Denote ζ = y/x and assume gener- ically that xy ≠ 0. Then the Secant map S induces a map Ŝ∞ ∶ `∞ → `∞ given by (10) Ŝ∞ [1 ∶ ζ ∶ 0] = [1 ∶ ϕk (ζ) ∶ 0] . Moreover the points {[1 ∶ 0 ∶ 0], [1 ∶ 1 ∶ 0]} are indeterminacy points of the secant map at infinity. Proof. The restriction of the Secant map (9) at infinity, that is with {z = 0}, under the generic condition xy ≠ 0, writes as ⎡ x (xk−1 − y k−1 ) ⎤ ⎢ ⎥ Ŝ∞ [x ∶ y ∶ 0] = [xk − y k ∶ (xk − xy k−1 ) ∶ 0] = ⎢1 ∶ ∶ 0⎥ . ⎢ (xk − yk ) ⎥ ⎣ ⎦ If ζ = y/x and we consider ϕk as defined in (5), we immediately conclude (10). The indeter- minacy points at infinity follows from the previous remark.  From the previous lemma it follows that the dynamics of Ŝ∞ is governed by the dynamics of the map ϕk in R. Lemma 3.2. The following statements hold. 8 ANTONIO GARIJO AND XAVIER JARQUE ϕk ϕk (ζ) = ζ ϕk ϕk (ζ) = ζ τk −1 −1 ● ● ● ● ● ● ● ηk 1 ζ ηk 1 ζ (a) Graphic of ϕk for k even. (b) Graphic of ϕk for k odd. Figure 4. The qualitative graph of the auxiliary map ϕk in R depending on the parity of k. (a) If k is odd, the map ϕk has a unique fixed point ηk . Moreover, ηk ∈ (0, 1) and it is a global attractor, i.e., ϕnk (ζ) → ηk as n → ∞ for all ζ ∈ R. (b) If k is even, the map ϕk has exactly two fixed points. One, ηk ∈ (0, 1), is attracting. The other τk < −1 is repelling. Moreover, ϕnk (ζ) → ηk as n → ∞ for all ζ ∈ R∖∆∪{τk }, where ∆ is defined in (6). Proof. We first prove (b). Clearly the map ϕk (ζ) has a unique vertical asymptote at ζ = −1, and a horizontal asymptote at y = 0 since ϕk (ζ) → 0± as ζ → ±∞. Easy computations show ζ k−2 ϕ′k (ζ) = − ψk (ζ), (xk − 1)2 (11) where ψk (ζ) = (ζ k − [k(ζ − 1) + 1]). We have that sign (ϕ′k (ζ)) = −sign (ψk (ζ)) since k is an even number. But observe that ψk (ζ) measures the difference from the graph of the function y = ζ k and the graph of the function y = k(ζ − 1) + 1 which is precisely the tangent line to the graph of y = ζ k at the point ζ = 1. Hence ψk (ζ) ≥ 0 for all ζ ∈ R and ψk (ζ) = 0 if and only if ζ = 1. In particular this implies that ϕk is strictly decreasing in its domain of definition. All together prove that ϕk has exactly two fixed points: τk < −1 and ηk ∈ (1/2, 1) (notice that ϕk (1/2) > 1/2 and ϕk (1) = (k − 1)/k < 1). See Figure 5(b). We claim that the positive fixed point ηk is attracting by showing that ∣ϕ′k (ηk )∣ < 1, and the negative fixed point τk is repelling by showing that ∣ϕ′k (τk )∣ > 1. We consider each case in turn. To see this we notice that ϕ′k (ζ) writes as ζ k−2 (ζ k−2 + 2ζ k−3 + . . . + (k − 2)ζ + k − 1) ϕ′k (ζ) = − . (ζ k−1 + xk−2 + . . . + ζ + 1) 2 DYNAMICS OF THE SECANT MAP NEAR INFINITY 9 Doing some computations we get ∑j=4 k+1 (j − 3)ζ 2k−j + (k − 1)ζ k−2 ϕ′k (ζ) =− , j=4 (j − 1)ζ ψ(ζ) + ∑k+1 2k−j + (k − 1)ζ k−2 where ψ(ζ) > 0 for all ζ > 0 (in fact it is a polynomial with positive coefficients). Accordingly ∣ϕ′k (ζ)∣ < 1 for all m > 0 since j − 3 < j − 1 and so the positive fixed point ηk of ϕk is attracting. The rest of the statement of (a) follows easily. To finish, we prove (b) by showing that τk < −1 is repelling since the other statements follow easily. Observe that ϕ′k (ζ) = [(k − 1)xk−2 ζ k − 1) − kζ k−1 (ζ k−1 − 1)] 1 (ζ k − 1) 2 and since τk is a fixed point of ϕk we conclude that τkk−1 −1 = τk (τkk − 1). So some computations give τ k−2 ϕ′k (τk ) = − kk [k(τk2 − 1) + 1] . τk − 1 Since τk < −1 and k is even it is clear that ϕ′k (τk ) < 0. We claim that ϕ′k (τk ) < −1 and so τk is a repelling fixed point of ϕk . Indeed −ϕ′k (τk ) > k + − 2 > 1. 1 k 2 τk τk We secondly assume that k is an odd number. We have that ϕk (ζ) → 0± as x → ±∞ but now ϕk (−1) = 0. See Figure 5 (b). In a similar way as in the even case ϕk (ζ) is a decreasing map for ζ ≥ 0 since ϕ′k (ζ) < 0 for all value of x ≥ 0. Hence, the map ϕk has a positive fixed point ηk in the interval (0, 1) since ϕk (0) = 1 and ϕk (1) = (k−1)/k. Using the same arguments as in the even case we conclude that ηk is an attracting fixed point. Finally, we claim that ηk is the unique fixed point of ϕk when k is an odd number. From (5) have that ϕk (ζ) = ζ writes as ζ k + ζ k−1 − 1 = 0 and it is easy to see that the polynomial qk (x) = ζ k + ζ k−1 − 1 has a unique real root when k is an odd number, since qk has a local minimum at 0 with qk (0) < 0 and a local maximum at −(k − 1)/k with k−1 k−1 k k − 1 k−1 (k − 1)k−1 qk (− ) = −( ) +( ) −1= − 1 < 0. k k k kk  In Table 1 we compute numerically the fixed points of ϕk for several values of k. Combining Lemmas 3.1 and 3.2 we obtain Theorem A. Proof of Theorem A. The statement of Theorem A is basically a direct consequence of Lem- mas 3.1 and 3.2. We only notice that in Theorem A(a) we exclude the point ζ = 0 since in Lemma 3.1 we had assumed xy ≠ 0, and we exclude ζ = −1 since ϕ(−1) = 0.  4. Further results and the secant map on CP2 We start this section by studding the the local dynamics of [1 ∶ ηk ∶ 0] and [1 ∶ τk ∶ 0] as fixed points of Ŝ in RP2 (that is, not only restricted to the line at infinity). We first write 10 ANTONIO GARIJO AND XAVIER JARQUE k attracting fixed point ηk of ϕk repelling fixed point τk of ϕk 2 ≈ 0,61803399 ≈ -1,61803399 3 ≈ 0,75487766 —— 4 ≈ 0,81917251 ≈ -1,38027757 5 ≈ 0,85667488 —— 6 ≈ 0,88127146 ≈ -1,28519903 7 ≈ 0,89865371 —— 8 ≈ 0,91159235 ≈ -1,23205463 9 ≈ 0,92159932 —— ⋯ ⋯ ⋯ 20 ≈ 0.9650705 ≈ -1,1186991 ⋯ ⋯ ⋯ 50 ≈ 0.9860941 ≈ -1.05933705 Table 1. Fixed points of the map S̃ on `∞ . the expression Ŝ in the x = 1 homogeneous coordinate system. We have Ŝ[1 ∶ y ∶ z] = [y (q (1, z) − q (y, z)) ∶ y q (1, z) − q (y, z) ∶ z (q (1, z) − q (y, z))] y q (1, z) − q (y, z) z (q (1, z) − q (y, z)) = [1 ∶ ∶ ] (12) y (q (1, z) − q (y, z)) y (q (1, z) − q (y, z)) (1 − y k−1 ) + O(z) z = [1 ∶ ∶ ] (1 − y k ) + O(z) y Using the notation (y, z) ↔ [1 ∶ y ∶ z] we might rewrite Ŝ as a map G acting as follows (1 − y k−1 ) + O(z) z (y, z) → G(y, z) = ( , ) = (G1 (y, z), G2 (y, z)) . (1 − y k ) + O(z) y The differential of G at any point of the form (ξ, 0) is a triangular matrix of the form ϕ′k (ξ) ⋆ (13) DG(ξ, 0) = ( ) 0 1/ξ We conclude in the following statement the local behviour of the real fixed points of Ŝ in `∞ . Lemma 4.1. The fixed points (ηk , 0) and (τk , 0) (only for k even), are of saddle type. Proof. We know from Lemma 3.2 that, on the one hand, (ηk , 0) is such that 0 < ηk < 1 and is an attracting fixed point −1 < ϕ′k (ηk ) < 0, and on the other hand, (τk , 0) is such that τk < −1 and is a repelling fixed point ϕ′k (τk ) < −1. So the lemma follows from (13).  The previous lemma complete the picture of the local dynamics of the infinite real fixed points of Ŝ. However, in CP2 , the equation ϕk (ζ) = ζ has k solutions, and only one or two of them (depending on the parity of k) have been already studied. We finish this section by studding the local dynamics around the non real infinite fixed points of Ŝ. Clearly (9) and (12) correspond to the expressions of Ŝ in CP2 , just taking the variables {x, y, z} as complex. So, the infinity of CP2 is now a Riemann sphere, S2 = Ĉ, and DYNAMICS OF THE SECANT MAP NEAR INFINITY 11 ● ● η3 η6 (a) (b) Figure 5. The dynamical planes of ϕk for k = 3 (left) and k = 6 (right). In or- ange (gradient orange-to-yellow) we draw the completely invariant immediate basin of attraction of ηk , k = 3, 6. The points in yellow (intersection of all gradient-orange) correspond to the Julia set which we conjecture it is a Cantor set. So, the Fatou set has a unique component, A⋆ (ηk ), the immediate basin of attraction of the point ηk , and all fixed points different from ηk are repelling. the dynamics of Ŝ∞ is governed by the complex rational map ϕk ∶ Ĉ → Ĉ as defined in (5) but now considering ζ ∈ C. Let R be a rational map. We say that a point z ∈ F(R), where F(R) denotes the Fatou set of R, if {f n ∣U }n≥0 is a normal family for some sufficiently small enigborhood U of z. In contrast, the Julia set is defined as J (R) = Ĉ ∖ F(R). The Fatou set is open (so, the Julia set is closed) and its connected components are called Fatou components. One can easily prove that the immediate basin of the attracting fixed point (as well as its eventual preimages, if any) are Fatou components. See [Mil06] for an excellent reference on complex rational iteration. In our setting for each k ≥ 3, the point ηk is an attracting fixed point. We denote by A⋆ (ηk ) its immediate basin of attraction. We conjecture that all critical points belong to A⋆ (ηk ) and then by Theorem 9.8.1 in [Bea91] the Julia set of ϕk is a Cantor set. Consequently all periodic points (of any period) different from ηk are repelling. In what follows we assume that all fixed points different from ηk are repelling as fixed points of Ŝ∞ in Ĉ. To study their local character as fixed points of Ŝ in CP2 , according to (13), we need to know if their modulus is greater or smaller than 1. A direct computation shows that for k = 3 the two fixed points different from η3 has modulus bigger than 1 (and so they are repelling fixed points in CP2 ) while for k ≥ 4 there are fixed points of modulus greater and smaller than 1, and hence for k ≥ 4 the fixed points can be saddle or repelling fixed points. 12 ANTONIO GARIJO AND XAVIER JARQUE 5. Conclusions Theorem A describes the dynamics of Ŝ∞ as a map of `∞ ⊂ RP2 , and it points out some differences depending on the parity of k (we illustrate these behaviours in Figures 2(b) and (d)). The first consideration is that Ŝ∞ always has an attracting fixed point [1 ∶ ηk ∶ 0] which basically trapped all the dynamics on the line at infinity; that is, except for a countable discrete set of points on `∞ , all initial conditions converge to [1 ∶ ηk ∶ 0]. It is worth to notice that this fixed point has a direct link with the two external roots of p; the largest and the smallest root of p which have wide sectors inside the immediate basin with [1 ∶ ηk ∶ 0] on their (infinite) boundaries. Notice also that diametrical opposite points in the circle correspond to the same homogenous coordinate. Secondly, we proved that the dynamics of Ŝ∞ is governed by the dynamics of the real map ϕk . We restrict now to k even (see Figure 5(a)). Clearly x = −1 is a vertical asymptote and y = 0 is an horizontal asymptote. In Lemma 3.2 we proved that ϕk has one attractive fixed point ηk , and one repelling fixed point τk < −1. Except for the preimages of x = −1 and the point x = τk any other point tends to τk under iteration. How this translates to the dynamics of Ŝ∞ ? Outside the points [1 ∶ 0 ∶ 0], [0 ∶ 1 ∶ 0] and [1 ∶ −1 ∶ 0] the map Sˆp reflects trivially the dynamics of ϕk , so every orbit converge to [1 ∶ ηk ∶ 0]. In fact [1 ∶ 0 ∶ 0] is the endpoint the horizontal lines Lj = {(x, αj ), x ∈ R, j = 0, . . . n} and it is an exercise to prove that, at least for large values of x, Lj ∈ A (αj ). This explains why we see in Figure 2(b) all colours (corresponding to all roots of p) landing at this point in `∞ . Now, since y = 0 is an horizontal asymptote for ϕk and Sˆp ([0 ∶ 1 ∶ 0]) = [1 ∶ 0 ∶ 0] we see a copy of the previous structure in the vertical direction. Thus, it only remains to discuss the point [1 ∶ −1 ∶ 0]. Certainly Sˆp has a pole at this point since ϕ(−1) = ∞ but it is possible to redefine Ŝ[1 ∶ −1 ∶ 0] ∶= [1 ∶ 1 ∶ 0] (we just use that lim S(x, −x) = [1 ∶ 1 ∶ 0] x→∞ and this is the reason why in Figure 2(b) we see nothing but red in a neighbourhood of [1 ∶ −1 ∶ 0]. If k is odd (see Figure 5(b) and Figure 2(d)), since lim ϕk (x) = 0 and ϕk (−1) = 0 x→±∞ the point [1 ∶ 0 ∶ 0] has two preimages: [0 ∶ 1 ∶ 0] and [1 ∶ −1 ∶ 0]. So, using the same arguments as above, we see all colours landing at [1 ∶ 0 ∶ 0] and in each of the two preimages. Since ϕnk (x) = ηk , x ≠ R ∖ {−1} we easily conclude that Sˆn ([1 ∶ x ∶ 0]) → [1 ∶ ηk ∶ 0] as n → ∞, for all x ∈ R ∖ {0, −1}. References [Bea91] Alan F. Beardon. Iteration of rational functions, volume 132 of Graduate Texts in Mathematics. Springer-Verlag, New York, NY, 1991. [BF19] Eric Bedford and Paul Frigge. The secant method for root finding, viewed as a dynamical system. Dolomites Research Notes on Approximation, to appear, 2019. [BFJK14] Krzysztof Barański, Núria Fagella, Xavier Jarque, and Boguslawa Karpińska. On the connectivity of the Julia sets of meromorphic functions. Invent. Math., 198(3):591–636, 2014. [BFJK18] Krzysztof Barański, Núria Fagella, Xavier Jarque, and Bogusl awa Karpińska. Connectivity of Julia sets of Newton maps: a unified approach. Rev. Mat. Iberoam., 34(3):1211–1228, 2018. DYNAMICS OF THE SECANT MAP NEAR INFINITY 13 [Cay79a] Arthur Cayley. Applications of the Newton-Fourier Method to an imaginary root of an equation. Quat. J. of Pure and App. Math., 16:179–185, 1879. [Cay79b] Arthur Cayley. The Newton-Fourier Method imaginary problem. Am. J. of Math., 2:97, 1879. [Cay80] Arthur Cayley. On the Newton-Fourier imaginary problem. Proc. Camb. Phil. Soc., 3:231–232, 1880. [GJ19] Antonio Garijo and Xavier Jarque. Global dynamics of the real secant method. Nonlinearity, 32(11):4557–4578, 2019. [GJ20] Antonio Garijo and Xavier Jarque. The secant map applied to a real polynomial with multiple roots. To appear in Discrete and Continuous Dynamical Systems, 2020. [HSS01] John Hubbard, Dierk Schleicher, and Scott Sutherland. How to find all roots of complex polynomials by Newton’s method. Invent. Math., 146(1):1–33, 2001. [Mil06] John Milnor. Dynamics in one complex variable, volume 160 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, third edition, 2006. [Shi09] Mitsuhiro Shishikura. The connectivity of the Julia set and fixed points. In Complex dynamics, pages 257–276. A K Peters, Wellesley, MA, 2009. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Catalonia. Email address: antonio.garijo@urv.cat Departament de Matemàtiques i Informàtica at Universitat de Barelona and Barcelona Graduate School of Mathematics, 08007 Barcelona, Catalonia. Email address: xavier.jarque@ub.edu