Fractals, Vol. 27, No. 6 (2019) 1950099 (22 pages) c World Scientific Publishing Company DOI: 10.1142/S0218348X19500993 CONJECTURES ABOUT SIMPLE DYNAMICS FOR SOME REAL NEWTON MAPS ON R2 ROBERTO DE LEO Howard University, Washington DC 20059, USA Received December 1, 2018 Accepted May 28, 2019 Published October 11, 2019 In memory of my father Giovanni (1942–2019), that in the good old days encouraged me to learn Basic on our Commodore 64. Abstract We collect from several sources some of the most important results on the forward and back- Fractals 2019.27. Downloaded from www.worldscientific.com ward limits of points under real and complex rational functions, and in particular real and complex Newton maps, in one variable and we provide numerical evidence that the dynamics 2 2 of Newton maps Nf associated to real polynomial maps f : R → R with no complex roots has a complexity comparable with that of complex Newton maps in one variable. In particular such a map Nf has no wandering domain, almost every point under Nf is asymptotic to a fixed point and there is some non-empty open set of points whose α-limit equals the set of non-regular points of the Julia set of Nf . The first two points were proved by B. Barna in the real one-dimensional case. by UNIVERSITY OF TORONTO on 01/03/20. Re-use and distribution is strictly not permitted, except for Open Access articles. Keywords: Newton’s Method; Barna’s Theorem; Discrete Dynamical Systems; Attractors; Repellors; Iterated Function Systems. 1. INTRODUCTION of the points of M under f. Among the simplest One of the most natural ways to understand the things that can happen is that there is some finite behavior of a continuous surjective map f of a number of attracting fixed points ci such that the compact manifold M into itself is studying the sequence of iterates {f n(x)} converges to one of asymptotics of the forward and backward orbits them for almost all x ∈ M (with respect to any 1950099-1 R. De Leo measure equivalent to the Lebesgue measure on providing numerical evidence that Newton’s maps each chart) and that, again for some full mea- on the real plane relative to generic polynomials sure set, the sets f −n(x) converge, in some suitable with only real roots are (weakly) plain. sense, to the set of points whose forward iterates do not converge to any ci. In other words, the action of f on M is, asymptotically, to thicken points 2. PRELIMINARIES near the ci while, at the same time, thinning them The following concepts are central for this paper. out near the boundaries of the basins of attrac- tion of the ci. We call functions with such behavior Definition 1. Let (M,µ) be a compact manifold plain. with a measure µ belonging to the Lebesgue mea- While the large diversity and complexity of sure class, namely a measure equivalent to the behaviors of continuous maps of a manifold M into Lebesgue measure on any chart, and let f be a itself suggests that in the general case the situation surjective continuous map of M into itself. The is much more complicated, it was a surprising dis- ω-limit of a point x ∈ M under f is the (closed) covery that the same is true even in case of very ele- set of the accumulation points of its forward orbit mentary maps such as quadratic polynomials in one {x,f(x),f(f(x)),...},namely variable (e.g. the logistic map) or piecewise linear m polynomials in one variable (e.g. the tent map) — ωf (x)= {f (x)}, see Refs. 1–4 and the references therein for a large n≥0 m≥n panorama of the old and recent advances in this while its α-limit is the (closed) set of the accumula- field. tion points of the sequence of preimages of x under Since being plain does not seem frequent among f,namely continuous functions, it is particularly important singling out properties that identify families of func- −m αf (x)= {f (x)}. tions that behave so nicely under iteration. A large n≥0 m≥n source of them is given by the rational maps com- ing from complex Newton’s method. Consider, for The ω-andα-limits of a set are defined similarly. instance, the case of the complex polynomial f(z)= The forward (respectively, backward) basin Ff (C) 3 1 1 z −1, whose Newton map Nf : CP → CP is given (respectively, Bf (C)) under f of a closed invariant 2z3+1 subset C ⊂ M is the set of all x ∈ M such that by Nf (z)= 3z2 .ItiswellknownthatNf has exactly three attractors, the cubic roots of the unity, ωf (x) ⊂ C (respectively, αf (x) ⊂ C). Following Milnor,10 we say that a closed subset C ⊂ M is an and one repellor, namely the Julia set of Nf (see attractor (respectively, repellor)forf if: Fig. 1, top), which means that Nf is plain. Note Fractals 2019.27. Downloaded from www.worldscientific.com that the situation can get more complicated even (1) Ff (C) (respectively, Bf (C)) has strictly posi- with different polynomials of same degree: as it was tive measure; 5 shown first numerically by Sullivan et al., in the (2) there is no closed subset C′ ⊂ C such that space of all complex cubic polynomials there is a Ff (C) (respectively, Bf (C)) coincides with set of non-zero Lebesgue measure for which there ′ ′ Ff (C ) (respectively, Bf (C )) up to a null set. exist attracting k-cycles, k ≥ 2, and, for each of these polynomials, the basin of attraction of such Finally, we say that f is plain if it has a finite num- attracting cycle has measure larger than zero (see ber of attracting fixed points ci, i =1,...,N,so by UNIVERSITY OF TORONTO on 01/03/20. Re-use and distribution is strictly not permitted, except for Open Access articles. Fig. 2, top). that: While the dynamics of Newton maps on the com- N plex line has been deeply and thoroughly studied (i) i=1 Ff (ci)=M\J is a full measure set; over the last 40 years, especially in connection with (ii) the set of x ∈ M such that αf (x)=J is a full the general problem of the dynamics of complex measure set. rational maps in one variable initiated exactly 100 If (ii) holds at least for a set of positive measure, 6 7–9 years ago by Julia and Fatou, in comparison then we say that f is weakly plain. almost nothing has been done in the more gen- eral case of Newton maps on the real plane. The Remark 1. Conditions (i) and (ii) imply that a main aim of this paper is to attract the attention of plain map cannot have any other attractor/repellor the dynamical systems community to this topic by besides the ci and J. 1950099-2 Conjectures About Simple Dynamics for Some Real Newton Maps on R2 Remark 2. While ω-limits of discrete systems have Theorem 1 (Refs. 6 and 7). Let f : CP1 → CP1 been thoroughly studied, at least in one (real and be a rational map of degree larger than 1.Then: complex) dimension, relatively very little has been done to date for α-limits (see Ref. 11 for a discussion (1) Ff contains all basins of attractions of f; on this topic). (2) Both Jf and Ff are forward and backward invariant and Jf is the smallest closed set with Throughout this paper we will endow all mani- more than two points with such property; folds M,asabove,withameasureµ belonging to (3) Jf is a perfect set; the Lebesgue measure class. Notice that all mea- (4) Jf has interior points if and only if Ff = ∅; sures within the Lebesgue measure class of M have (5) Jf = Jf n for all n ∈ N; the very same null sets so that, since all our state- (6) Jf is the closure of all repelling cycles of f; ments relative to measures of sets are about whether (7) ωf (z)=Jf for a generic point z ∈ Jf ; some set or its complement have zero or positive (8) αf (z)=Jf for every point z ∈ Jf ; measure, our results do not depend on the particu- (9) ∂Ff (γ)=Jf for every attracting periodic lar measure used within this class. By a slight abuse orbit γ; of notation, we will refer to such a measure as the (10) The dynamics of the restriction of f to its Julia Lebesgue measure on M. The Hausdorff measures set is highly sensitive to the initial conditions, induced on the spheres Sn by their round metric, namely f|Jf is chaotic. namely the angular distance between points, are an 1 example of such measures. Remark 3. The possibility that Jf = CP in point Finite attractors and repellors play a major role (4) above does take place. Two well-known exam- in this theory. ples of functions with empty Fatou set are the 2 2 12 (z +1) Latt`es example p(z)= 2 , related to the the- Definition 2. A periodic orbit (or k-cycle) γ is a 4z(z −1) non-empty finite set of k points minimally invariant ory of Elliptic functions (see Ref. 13), and q(z)= (z−2)2 under f, namely that cannot be decomposed into z2 (see Ref. 13 and Corollary 6.2.4 in Ref. 14). 1 the disjoint union of smaller invariant sets. A 1-cycle In general, Ff = ∅ if and only if ωf (z)=CP for is also called a fixed point.