arXiv subjects
Publications and source records attributed to Sarah Koch.
Recently, Noytaptim and Petsche proved that the only totally real parameters $c\in \overline{\mathbb Q}$ for which $f_c(z):=z^2+c$ is postcritically finite are $0$, $-1$ and $-2$. In this note, we show that the only totally real parameters $c\in \overline{\mathbb Q}$ for which $f_c$ has a parabolic cycle are $\frac14$, $-\frac34$, $-\frac54$ and $-\frac74$.
Given a rational map $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ on the Riemann sphere, we define $\mathrm{Deck}(f)$ to be the group of M\"obius transformations $\mu$ satisfying $f \circ \mu = f$. In this note, we consider the groups $\mathrm{Deck}(f^k)$, where $f$ is a \emph{bicritical} rational map (that is, a rational map with exactly two critical points) and $f^k$ denotes the $k$th iterate of $f$. In particular, we give a complete description of which groups (up to isomorphism) arise as the groups $\mathrm{Deck}(f^k)$ for bicritical rational maps $f$.
Let $f$ be a degree $d$ bicritical rational map with critical point set $\mathcal{C}_f$ and critical value set $\mathcal{V}_f$. Using the group $\textrm{Deck}(f^k)$ of deck transformations of $f^k$, we show that if $g$ is a bicritical rational map which shares an iterate with $f$ then $\mathcal{C}_f = \mathcal{C}_g$ and $\mathcal{V}_f = \mathcal{V}_g$. Using this, we show that if two bicritical rational maps of even degree $d$ share an iterate then they share a second iterate, and both maps belong to the symmetry locus of degree $d$ bicritical rational maps.
It is well known that the dynamical behavior of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is governed by the forward orbits of the critical points of $f$. The map $f$ is said to be postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle of $f$. We encode the orbits of the critical points of $f$ with a finite directed graph called a ramification portrait. In this article, we study which graphs arise as ramification portraits. We prove that every abstract polynomial portrait is realized as the ramification portrait of a postcritically finite polynomial, and classify which abstract polynomial portraits can only be realized by unobstructed maps.
We show that the set of conjugacy classes of cubic polynomials with a prefixed critical point, of preperiod $k\geq 1$, is an irreducible algebraic curve. We also establish an analogous result for quadratic rational maps. We then study a closely related question concerning the irreducibility (over $\mathbb Q$) of the set of conjugacy classes of unicritical polynomials, of degree $D\geq 2$, with a preperiodic critical point. Our proofs are purely algebraic.
Let $f:\hat{\mathbb C}\to \hat{\mathbb C}$ be a postcritically finite rational map. Let $\mathcal Q(\hat{\mathbb C})$ be the space of meromorphic quadratic differentials on $ \hat{\mathbb C}$ with simple poles. We study the set of eigenvalues of the pushforward operator $f_*:\mathcal Q(\hat{\mathbb C})\to \mathcal Q(\hat{\mathbb C})$. In particular, we show that when $f:\mathbb C \to \mathbb C$ is a unicritical polynomial of degree $D$ with periodic critical point, the eigenvalues of $f_*:\mathcal Q(\hat{\mathbb C})\to \mathcal Q(\hat{\mathbb C})$ are contained in the annulus $\bigl\{\frac{1}{4D}<|λ|<1\bigr\}$ and belong to $\frac{1}{D} \mathbb U$ where $\mathbb U$ is the group of algebraic units.
We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena.
Let $f:(\mathbb{P}^1,P)\to(\mathbb{P}^1,P)$ be a postcritically finite rational map with postcritical set $P$. William Thurston showed that $f$ induces a holomorphic pullback map $σ_f:\mathcal{T}_P\to\mathcal{T}_P$ on the Teichmüller space ${\mathcal T}_P:=\mathrm{Teich}(\mathbb{P}^1,P)$. If $f$ is not a flexible Lattès map, Thurston proved that $σ_f$ has a unique fixed point. In his PhD thesis, Adam Epstein generalized Thurston's ideas and defined a deformation space associated to a rational map $f:(\mathbb{P}^1,A)\to (\mathbb{P}^1,B)$ where $A \subseteq B$, allowing for maps $f$ which are not necessarily postcritically finite. By definition, the deformation space $\mathrm{Def}_B^A(f)\subseteq \mathcal{T}_B$ is the locus where the pullback map $σ_f:\mathcal{T}_B\to\mathcal{T}_A$ and the forgetful map $σ_A^B:\mathcal{T}_B\to\mathcal{T}_A$ agree. Using purely local arguments, Epstein showed that $\mathrm{Def}_B^A(f)$ is a smooth analytic submanifold of $\mathcal{T}_B$ of dimension $|B-A|$. In this article, we investigate the question of whether $\mathrm{Def}_B^A(f)$ is connected. We exhibit a family of quadratic rational maps for which the associated deformation spaces are disconnected; in fact, each has infinitely many components.
The dynamical degrees of a rational map $f:X\dashrightarrow X$ are fundamental invariants describing the rate of growth of the action of iterates of $f$ on the cohomology of $X$. When $f$ has nonempty indeterminacy set, these quantities can be very difficult to determine. We study rational maps $f:X^N\dashrightarrow X^N$, where $X^N$ is isomorphic to the Deligne-Mumford compactification $\overline {\mathcal M}_{0,N+3}$. We exploit the stratified structure of $X^N$ to provide new examples of rational maps, in arbitrary dimension, for which the action on cohomology behaves functorially under iteration. From this, all dynamical degrees can be readily computed (given enough book-keeping and computing time). In this article, we explicitly compute all of the dynamical degrees for all such maps $f:X^N\dashrightarrow X^N$, where $\mathrm{dim}(X^N)\leq 3$ and the first dynamical degrees for the mappings where $\mathrm{dim}(X^N)\leq 5$. These examples naturally arise in the setting of Thurston's topological characterization of rational maps.
Let $f:S^2\to S^2$ be an orientation-preserving branched covering map of degree $d\geq 2$, and let $Σ$ be an oriented Jordan curve passing through the critical values of $f$. Then $Γ:=f^{-1}(Σ)$ is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs and showed that they combinatorially characterize all such $Γ$, where $f$ has $2d-2$ distinct critical values. We give a detailed account of this discussion, along with some examples and an appendix about Hurwitz numbers.
In 1985, Barnsley and Harrington defined a ``Mandelbrot Set'' $\mathcal{M}$ 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 $x \mapsto zx$ and $x \mapsto z(x-1)+1$ is connected. Equivalently, $\mathcal{M}$ is the closure of the set of roots of polynomials with coefficients in $\lbrace -1,0,1 \rbrace$. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small ``holes'' in $\mathcal{M}$, 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 $\mathcal{M}$, and use them to prove Bandt's Conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in $\mathcal{M}$.
In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of the moduli space. We put an analytic structure on this compact quotient and prove that with respect to this structure, it is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification.
Associated to a Thurston map $f: S^2 \to S^2$ with postcritical set $P$ are several different invariants obtained via pullback: a relation on the set of free homotopy classes of curves in $S^2- P$, a linear operator on the free $\R$-module generated by these homotopy classes of curves, a virtual endomorphism on the pure mapping class group, an analytic self-map of an associated Teichmueller space, and an analytic self-correspondence on an associated moduli space. Viewing all of these objects as invariants of $f$, we investigate harmonious relationships between their properties.
Let f: P^1 \to P^1 be a rational map with finite postcritical set P_f. Thurston showed that f induces a holomorphic map σ_f of the Teichmueller space T modelled on P_f to itself fixing the basepoint corresponding to the identity map (P^1, P_f) \to (P^1, P_f). We give explicit examples of such maps f showing that the following cases may occur: (1) the basepoint is an attracting fixed point, the image of σ_f is open and dense, and the map σ_f is a covering map onto its image; (2) the basepoint is a superattracting fixed point, σis surjective, and σis a ramified Galois covering, (3) σ_f is constant.
A well-known theorem of Böttcher asserts that an analytic germ f:(C,0)->(C,0) which has a superattracting fixed point at 0, more precisely of the form f(z) = az^k + o(z^k) for some a in C^*, is analytically conjugate to z->az^k by an analytic germ phi:(C,0)->(C,0) which is tangent to the identity at 0. In this article, we generalize this result to analytic maps of several complex variables.