COMBINATORICS AND TOPOLOGY OF THE SHIFT LOCUS LAURA DE MARCO Abstract. As studied by Blanchard, Devaney, and Keen in [BDK], closed loops in the shift locus (in the space of polynomials of degree d) induce automorphisms of the full one-sided d-shift. In this article, I describe how to compute the induced automorphism from the pictograph of a polynomial (introduced in [DP2]) for twist-induced loops. This article is an expanded version of my lecture notes from the conference in honor of Linda Keen's birthday, in October of 2010. Happy Birthday, Linda! 1. Introduction In [DP2], Kevin Pilgrim and I introduced the pictograph, a diagrammatic representation of the basin of infinity of a polynomial, with the aim of classifying topological conjugacy classes. The pictograph is almost a complete invariant for polynomials in the shift locus, those for which all critical points are attracted to 1. In the shift locus, the number of topological conjugacy classes with a given pictograph can be computed directly from the pictograph, and it is always finite. All polynomials in the shift locus are topologically conjugate on their Julia sets; in each degree d ≥ 2, they are conjugate to the one-sided shift on d symbols. In degrees d > 2, however, the conjugacies may fail to extend to the full complex plane. Indeed, there are infinitely many global topological conjugacy classes of polynomials in the shift locus, for each degree d > 2. In [DP1], Kevin and I looked at the way these topological conjugacy classes fit together within the moduli space of conformal conjugacy classes. For example, in degree 3, there is a locally finite simplicial tree that records how the (structurally stable) conjugacy classes are adjacent. The edges and vertices of the tree can be encoded by the pictographs of [DP2]. During my first presentation about pictographs, at the conference in honor of Bob De- vaney's birthday (Tossa de Mar, Spain, April 2008), Linda Keen asked: what is the relation between your combinatorics and the automorphisms of the shift induced by loops in the shift locus? She referred to her work with Blanchard and Devaney in [BDK], where they proved that the fundamental group of the shift locus surjects onto the group of automorphisms of the one-sided shift; see x2 below. My lecture at the conference in honor of Linda's birthday (New York, NY, October 2010) was devoted to this relation. This article is an expanded version of the notes from my lecture. In this article, I will describe the relation between topological conjugacy classes in the shift locus and automorphisms of the shift as studied in [BDK], and I pose a few problems. The loops in the shift locus constructed in [BDK] are produced via twisting deformations of Date: December 15, 2011. 2010 Mathematics Subject Classification. Primary 37F10, 37F20. 2 LAURA DE MARCO polynomials. In general, we can determine the action of a twist-induced shift automorphism from the data of the pictograph; see x3. The construction of abstract pictographs with interesting combinatorial properties leads to loops inducing shift automorphisms of varying orders. In degree 3, an explicit connection between shift automorphisms and the pictographs may be viewed as a \top-down" approach to understanding the organization and structure of stable conjugacy classes. This is to be contrasted with the \bottom-up" approach of [DS], where we built the tree of conjugacy classes in degree 3, starting with the Branner-Hubbard tableaux of [BH2], enumerating all of the associated pictographs, and finally counting the corresponding number of conjugacy classes. Details for cubic polynomials are given in x4. Acknowledgement. I would like to thank Paul Blanchard and Bob Devaney for some useful and inspiring conversations. My research is supported by the National Science Foun- dation and the Sloan Foundation. 2. The space of polynomials and the shift locus Following [BDK, BH1], it is convenient to parametrize the space of polynomials by their coefficients. We let Pd denote the space of monic and centered polynomials; i.e. polynomials of the form d d−2 f(z) = z + a2z + ··· + ad d−1 d−1 for complex coefficients (a2; : : : ; ad) 2 C , so that Pd ' C . Recall that the filled Julia set of a polynomial f is the compact subset of points with bounded orbit, n K(f) = fz 2 C : sup jf (z)j < 1g; n and its complement is the open, connected basin of infinity, n X(f) = fz 2 C : f (z) ! 1g = C n K(f): The shift locus in Pd consists of polynomials for which all critical points lie in the basin of infinity: 0 Sd = ff 2 Pd : c 2 X(f) for all f (c) = 0g: The terminology comes from the following well-known fact (see e.g. [Bl]): Theorem 2.1. If f 2 Sd, then K(f) is homeomorphic to a Cantor set, and fjK(f) is topologically conjugate to the one-sided shift map on d symbols. We let N Σd = f0; 1; : : : ; d − 1g denote the shift space, the space of half-infinite sequences on an alphabet of d letters, with its natural product topology making it homeomorphic to a Cantor set. The shift map σ :Σd ! Σd acts by cutting off the first letter of any sequence, σ(x1; x2; x3;:::) = (x2; x3;:::): It has degree d. COMBINATORICS AND TOPOLOGY OF THE SHIFT LOCUS 3 The polynomials in the shift locus are J-stable, in the language of McMullen and Sullivan [McS]. That is, throughout Sd, the Julia set of a polynomial f moves holomorphically, via a motion inducing a conjugacy on K(f); see also [Mc]. Fixing a basepoint f0 2 Sd and a topological conjugacy (f0;K(f0)) ∼ (σ; Σd), any closed loop in Sd starting and ending at f0 will therefore induce an automorphism of the shift. That is, the loop induces a homeomorphism ' :Σd ! Σd that commutes with the action of σ. In this way, we obtain a well-defined homomorphism π1(Sd; f0) ! Aut(σ; Σd): As the shift locus is connected (see e.g. [DP3, Corollary 6.2] which states that the image of Sd in the moduli space is connected, and observe that there are polynomials in Sd with automorphism of the maximal order d − 1, so Sd itself is connected), this homomorphism is independent of the basepoint, up to conjugacy within Aut(σ; Σd). In the beautiful article [BDK], Paul Blanchard, Bob Devaney, and Linda Keen proved: Theorem 2.2. The homomorphism π1(Sd; f0) ! Aut(σ; Σd) is surjective in every degree d ≥ 2. To appreciate this statement, we need to better understand the structure of the group Aut(σ; Σd). First consider the case of d = 2. The space P2 is a copy of C, parametrized by 2 the family fc(z) = z + c with c 2 C. The shift locus is the complement of the compact and connected Mandelbrot set, and therefore π1(S2) ' Z. Fixing a basepoint c0 2 S2, and fixing a topological conjugacy (fc0 ;K(fc0 )) ∼ (σ; Σ2), it is easy to see that a loop around the Mandelbrot set will interchange the symbols 0 and 1. In fact, starting with c0 < −2, if you watch a movie of the Julia sets of fc as c goes along a loop around the Mandelbrot set, you will see the two sides of the Julia set (on either side of z = 0 on the real line) exchange places. A theorem of Hedlund states that Aut(σ; Σ2) ' Z=2Z acting by interchanging the two letters of the alphabet [He]. Thus, the generator of π1(S2; fc0 ) is sent to the generator of Aut(σ; Σ2). In higher degrees, the topology of Sd and the group Aut(σ; Σd) are significantly more complicated. Simultaneous with the work of Blanchard-Devaney-Keen, the authors Mike Boyle, John Franks, and Bruce Kitchens studied the structure of Aut(σ; Σd) in degrees d > 2 [BFK]. To give you a flavor of its complexity, one of the results in [BFK] states: Theorem 2.3. For each d > 2, the group Aut(σ; Σd) is infinitely generated by elements of finite order. For every integer of the form n1 n2 nk N = p1 p2 ··· pk with primes pi < d and positive integers ni, there exists an element in Aut(σ; Σd) with order N. Further, for d not prime, every element of finite order in Aut(σ; Σd) has order of this form. If d is prime, then an element of finite order may also have order d. 4 LAURA DE MARCO At the same time these results were obtained, Jonathan Ashley devised an algorithm to produce a list of elements of Aut(σ; Σd) called marker automorphisms, each of order 2. Together with the permutations of the d letters, these marker automorphisms generate all of Aut(σ; Σd), for any d > 2 [Ash]. A marker automorphism of the shift is an automorphism of the following type: given a finite word w in the alphabet of Σd (or given a finite set of finite words), and given a transposition (a b) interchanging two elements of the alphabet, the marker automorphism acts on a symbol sequence by interchanging a and b when they are found immediately preceding the word w. The word w is called the marker of the associated automorphism. The strategy of proof in [BDK] was to construct loops in Sd that induce each of the marker automorphisms. More will be said about these \Blanchard-Devaney-Keen loops" later. 3. Topological conjugacy, the pictograph, and shift automorphisms 3.1. Topological conjugacy classes. A fundamental problem in the study of dynamical systems is to classify the topological conjugacy classes.