ROOTS, SCHOTTKY SEMIGROUPS, AND A PROOF OF BANDTS CONJECTURE DANNY - PDF document

ROOTS, SCHOTTKY SEMIGROUPS, AND A PROOF OF BANDT’S CONJECTURE DANNY CALEGARI, SARAH KOCH, AND ALDEN WALKER Abstract. In 1985, Barnsley and Harrington defined a “Mandelbrot Set” M for pairs of similarities — this is the set of complex numbers z with 0 < | z | < 1 for which the limit set of the semigroup generated by the similarities x �→ zx and x �→ z ( x − 1) + 1 is connected. Equivalently, M is the closure of the set of roots of polynomials with coefficients in {− 1 , 0 , 1 } . Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small “holes” in M , and conjectured that these holes were genuine. These holes are very interesting, since they are “exotic” components of the space of (2 generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, and he conjectured that the interior points are dense away from the real axis. We introduce the technique of traps to construct and certify interior points of M , and use them to prove Bandt’s Conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in M . Contents 1. Introduction 1 2. Semigroups of similarities 4 3. Elementary estimates 11 4. Roots, polynomials, and power series with regular coefficients 13 5. Topology and geometry of the limit set 17 6. Limit sets of differences 23 7. Interior points in M 25 8. Holes in M 32 9. Infinitely many holes in M and renormalization 39 10. Whiskers 53 11. Holes in M 0 60 References 64 1. Introduction Consider the similarity transformations f, g : C → C given by f : x �→ zx and g : x �→ z ( x − 1) + 1 , where z ∈ D ∗ := { z ∈ C | 0 < | z | < 1 } . Because these maps are contractions, there is a nonempty compact attractor Λ z ⊆ C associated with the iterated function Date : October 30, 2014. 1

2 DANNY CALEGARI, SARAH KOCH, AND ALDEN WALKER system (or IFS) given by the pair { f, g } . The attractor Λ z coincides with the set of accumulation points of the forward orbit of any x ∈ C under the semigroup G z := � f, g � . In this article, we study the topology of certain subsets of the parameter space D ∗ for G z . The first set we consider is the connectedness locus , denoted by M ; that is, the set of parameters z for which Λ z is connected. Standard IFS arguments prove that the limit set Λ z is either connected, or it is a Cantor set (for details, see Lemma 5.2.1). The second subset of the parameter space we examine is related to the geometry of Λ z . For all values of the parameter z ∈ D ∗ , the map f fixes 0, and the map g fixes 1. As both of these maps are contracting by the same factor (in fact, by a factor of z ) around their respective fixed points, the limit set Λ z has a center of symmetry about the point 1 / 2 in the dynamical plane. The set M 0 is defined to be the set of parameters z for which Λ z contains the point 1 / 2. The sets M and M 0 have been studied by various mathematicians over the past 30 years: Barnsley-Harrington [2], Bousch [3, 4], Bandt [1], Solomyak [11, 12], Shmerkin-Solomyak [10], and Solomyak-Xu [13], to name a few. There is a profound and unexpected connection between the sets M and M 0 and the set of roots of power series with prescribed coefficients (see Section 4). In particular, M can be identified with the closure of the set of roots of polynomials with coefficients in {− 1 , 0 , 1 } (which are in D ∗ ), and M 0 can be identified with the closure of the set of roots of polynomials with coefficients in {− 1 , 1 } (which are in D ∗ ). Via this formulation, the set M 0 is related to roots of the minimal polynomials associated to the core entropy of real quadratic polynomials as defined by Thurston [14], and established by Tiozzo [15]. We further elaborate on the history of M and M 0 in Section 2.6. In [3] and [4], Bousch proved that the sets M and M 0 are connected and locally connected. However, the complement of M and the complement of M 0 are discon- nected . The complement of M and the complement of M 0 both contain a prominent central component (see Figure 2 and Figure 3). In 1985, Barnsley and Harrington numerically observed other connected components of the complement, or “holes” in M , and they conjectured that these holes are genuine. In 2002, Bandt rigorously established the existence of one hole in M . In Theorem 9.1.1, we prove that there are infinitely many holes in M . These “exotic holes” in M are quite interesting and somewhat mysterious; they appear to be very well-organized in parameter space, suggesting that there may be a combinatorial classification of them. We currently have found no such classification. 1.1. Statement of results. We prove that all of the connected components of D ∗ \ M are Schottky , in the sense that if z in D ∗ \ M , there is a topological disk D containing Λ z , so that f ( D ) ∩ g ( D ) = ∅ , and f ( D ) and g ( D ) are contained in the interior of D . Theorem 5.2.3 (Disconnected is Schottky). The semigroup G z has discon- nected Λ z if and only if G z is Schottky. To prove that these exotic components in the complement of M exist, we intro- duce the method of traps (see Section 7.1), which allows us to numerically certify that a parameter z ∈ M . This technique is different from Bandt’s proof of the existence of these exotic holes. In fact, the existence of a trap is an open condition,

ROOTS, SCHOTTKY SEMIGROUPS, AND A PROOF OF BANDT’S CONJECTURE 3 so if there is a trap for the parameter z ∈ D ∗ , then necessarily z ∈ int( M ). Traps therefore allow us to access the interior points of M . In [1], Bandt conjectured that the interior of M is dense away from M ∩ R (see Conjecture 2.6.3). In Theorem 7.2.7, we prove Bandt’s conjecture using traps. Theorem 7.2.7 (Interior is almost dense). The interior of M is dense away from the real axis; that is, M = int( M ) ∪ ( M ∩ R ) . Interestingly, the proof of Theorem 7.2.7 requires a complete characterization of the set of z ∈ M for which the limit set Λ z is convex. This is established in Lemma 7.2.3. In Section 9, we examine families of exotic holes in M which appear to spiral down and limit on a distinguished point z ∈ ∂ M (see Figure 20). Theorem 9.1.1 (Limit of holes). Let ω ∼ 0 . 371859 + 0 . 519411 i be the root of the polynomial 1 − 2 z + 2 z 2 − 2 z 5 + 2 z 8 with the given approximate value. Then (1) ω is in M , and M 0 ; in fact, the intersection of f Λ ω and g Λ ω is exactly the point 1 / 2; (2) there are points in the complement of M arbitrarily close to ω ; and (3) there are infinitely many rings of concentric loops in the interior of M which nest down to the point ω . Thus, M contains infinitely many holes which accumulate at the point ω . We continue Section 9 by generalizing the methods of Theorem 9.1.1. We define the notion of renormalization and limiting traps to show that at certain renormal- ization points z ∈ M , the set M is asymptotically similar to Γ z , where Γ z is the limit set of the 3 generator IFS x �→ z ( x + 1) − 1 x �→ zx x �→ z ( x − 1) + 1 . Previous results of Solomyak established this asymptotic similarity at certain ‘land- mark points’ in ∂ M . We reprove his results with a more algorithmic approach using traps, and as a consequence, we obtain “asymptotic interior.” Theorem 9.2.2 (Renormalizable traps). Suppose that ω is a renormalization point. There are constants A and B , depending only on ω , such that (1) If C ∈ ( A + B Γ ω ), then for all ǫ > 0, there is a C ′ such that | C − C ′ | < ǫ and for all sufficiently large n , there is a trap for ω + C ′ ω bn . (2) If f Λ z ∩ g Λ z is a single point, then there is δ > 0 such that for all C / ∈ ( A + B Γ ω ) with | C | < δ , the limit set for the parameter ω + Cω bn is disconnected for all sufficiently large n . In Section 11, we prove that the complement of M 0 is also disconnected by numerically certifying a loop in M 0 which bounds a component of the complement. Theorem 11.3 (Hole in M 0 ). There is a hole in M 0 . 1.2. Outline. In Section 2, we establish key definitions and survey some previous results about M and M 0 . In Section 3, we collect a few elementary estimates about the geometry of Λ z . In Section 4, we explore the connection the sets M and M 0 have with roots of power series with prescribed coefficients in a more general context involving regular languages. In Section 5, we establish some important