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Saeed Zakeri

Publications and source records attributed to Saeed Zakeri.

17 recordsLinked to original sources
arXiv2025

Lemon limbs of the cubic connectedness locus

We describe a primary limb structure in the connectedness locus of complex cubic polynomials, where the limbs are indexed by the periodic points of the doubling map $t \mapsto 2t \ (\operatorname{mod} {\mathbb Z})$. The main renormalization locus in each limb is parametrized by the product of a pair of (punctured) Mandelbrot sets. This parametrization is the inverse of the straightening map and can be thought of as a tuning operation that manufactures a unique cubic of a given combinatorics from a pair of quadratic hybrid classes.

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arXiv2024

Buff forms and invariant curves of near-parabolic maps

We introduce a general framework to study the local dynamics of near-parabolic maps using the meromorphic $1$-form introduced by X.~Buff. As a sample application of this setup, we prove the following tameness result on invariant curves of near-parabolic maps: Let $g(z)=λz+O(z^2)$ have a non-degenerate parabolic fixed point at $0$ with multiplier $λ$ a primitive $q$th root of unity, and let $γ: \, ]-\infty,0] \to {\mathbb D}(0,r)$ be a $g^{\circ q}$-invariant curve landing at $0$ in the sense that $g^{\circ q}(γ(t))=γ(t+1)$ and $\lim_{t \to -\infty} γ(t)=0$. Take a sequence $g_n(z)=λ_n z+O(z^2)$ with $|λ_n|\neq 1$ such that $g_n \to g$ uniformly on ${\mathbb D}(0,r)$ and suppose each $g_n$ admits a $g_n^{\circ q}$-invariant curve $γ_n: \, ]-\infty,0] \to {\mathbb C}$ such that $γ_n \to γ$ uniformly on the fundamental segment $[-1,0]$. If $λ_n^q \to 1$ non-tangentially, then $γ_n$ lands at a repelling periodic point near $0$, and $γ_n \to γ$ uniformly on $]-\infty,0]$. In the special case of polynomial maps, this proves Hausdorff continuity of external rays of a given periodic angle when the associated multipliers approach a root of unity non-tangentially.

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arXiv2021

Cyclic Permutations: Degrees and Combinatorial Types

This note will give an enumeration of $n$-cycles in the symmetric group ${\mathcal S}_n$ by their degree (also known as their cyclic descent number) and studies similar counting problems for the conjugacy classes of $n$-cycles under the action of the rotation subgroup of ${\mathcal S}_n$. This is achieved by relating such cycles to periodic orbits of an associated dynamical system acting on the circle. We also compute the mean and variance of the degree of a random $n$-cycle and show that its distribution is asymptotically normal as $n \to \infty$.

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arXiv2020

Periodic Points and Smooth Rays

Let $P: {\mathbb C} \to {\mathbb C}$ be a polynomial map with disconnected filled Julia set $K_P$ and let $z_0$ be a repelling or parabolic periodic point of $P$. We show that if the connected component of $K_P$ containing $z_0$ is non-degenerate, then $z_0$ is the landing point of at least one {\it smooth} external ray. The statement is optimal in the sense that all but one ray landing at $z_0$ may be broken.

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arXiv2019

On Combinatorial Types of Periodic Orbits of the Map $x \mapsto kx$ (mod $\mathbb Z$)

We study the combinatorial types of periodic orbits of the standard covering endomorphisms ${\mathbf m}_k(x)=k x \ (\text{mod} \ {\mathbb Z})$ of the circle for integers $k \geq 2$ and the frequency with which they occur. For any $q$-cycle $σ$ in the permutation group ${\mathcal S}_q$, we give a full description of the set of period $q$ orbits of ${\mathbf m}_k$ that realize $σ$ and in particular count how many such orbits there are. The description is based on an invariant called the "fixed point distribution" vector and is achieved by reducing the realization problem to finding the stationary state of an associated Markov chain. Our results generalize earlier work on the special case where $σ$ is a rotation cycle, and can be viewed as a missing combinatorial ingredient for a proper understanding of the dynamics of complex polynomial maps of degree $\geq 3$ and the structure of their parameter spaces.

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arXiv2019

On the correspondence of external rays under renormalization

Let $P$ be a monic polynomial of degree $D \geq 3$ whose filled Julia set $K_P$ has a non-degenerate periodic component $K$ of period $k \geq 1$ and renormalization degree $2 \leq d<D$. Let $I=I_K$ denote the set of angles $θ$ on the circle ${\mathbb T}={\mathbb R}/{\mathbb Z}$ for which the (smooth or broken) external ray $R^P_θ$ for $P$ accumulates on $\partial K$. We prove the following: $\bullet$ $I$ is a compact set of Hausdorff dimension $<1$ and there is an essentially unique degree $1$ monotone map $Π: I \to {\mathbb T}$ which semiconjugates $θ\mapsto D^k θ$ (mod 1) on $I$ to $θ\mapsto d θ$ (mod 1) on $\mathbb T$. $\bullet$ Any hybrid conjugacy $φ$ between a renormalization of $P^{\circ k}$ on a neighborhood of $K$ and a monic degree $d$ polynomial $Q$ induces a semiconjugacy $Π: I \to {\mathbb T}$ with the property that for every $θ\in I$ the external ray $R^P_θ$ has the same accumulation set as the curve $φ^{-1}(R^Q_{Π(θ)})$. In particular, $R^P_θ$ lands at $z \in \partial K$ if and only if $R^Q_{Π(θ)}$ lands at $φ(z) \in \partial K_Q$. $\bullet$ The ray correspondence established by the above result is finite-to-one. In fact, the cardinality of each fiber of $Π$ is $\leq D-d+2$, and the inequality is strict when the component $K$ has period $k=1$. Using a new type of quasiconformal surgery we construct a class of examples with $k=1$ for which the upper bound $D-d+1$ is realized and the set $I$ has isolated points.

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arXiv2013

A discreteness criterion for groups containing parabolic isometries

This note will prove a discreteness criterion for groups of orientation-preserving isometries of the hyperbolic space which contain a parabolic element. It can be viewed as a generalization of the well-known results of Shimizu-Leutbecher and Jorgensen in dimensions 2 and 3, and is closely related to Waterman's inequality in higher dimensions. Unlike his algebraic method, the argument presented here is geometric and yields an improved asymptotic bound.

math.GT↗
arXiv2013

On Margulis cusps of hyperbolic 4-manifolds

We study the geometry of the Margulis region associated with an irrational screw translation $g$ acting on the 4-dimensional real hyperbolic space. This is an invariant domain with the parabolic fixed point of $g$ on its boundary which plays the role of an invariant horoball for a translation in dimensions $\leq 3$. The boundary of the Margulis region is described in terms of a function $B_α: [0,\infty) \to {\mathbb R}$ which solely depends on the rotation angle $α\in {\mathbb R}/{\mathbb Z}$ of $g$. We obtain an asymptotically universal upper bound for $B_α(r)$ as $r \to \infty$ for arbitrary irrational $α$, as well as lower bounds when $α$ is Diophatine and the optimal bound when $α$ is of bounded type. We investigate the implications of these results for the geometry of Margulis cusps of hyperbolic 4-manifolds that correspond to irrational screw translations acting on the universal cover. Among other things, we prove bi-Lipschitz rigidity of these cusps.

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arXiv2013

Conformal Fitness and Uniformization of Holomorphically Moving Disks

Let $\{U_t \}_{t \in {\mathbb D}}$ be a family of topological disks on the Riemann sphere containing the origin 0 whose boundaries undergo a holomorphic motion over the unit disk $\mathbb D$. We study the question of when there exists a family of Riemann maps $g_t:({\mathbb D},0) \to (U_t,0)$ which depends holomorphically on the parameter $t$. We give five equivalent conditions which provide analytic, dynamical and measure-theoretic characterizations for the existence of the family $\{g_t \}_{t \in {\mathbb D}}$, and explore the consequences.

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arXiv2002

External rays and the real slice of the Mandelbrot set

This paper investigates the set of angles of the parameter rays which land on the real slice $[-2,1/4]$ of the Mandelbrot set. We prove that this set has zero length but Hausdorff dimension 1. We obtain the corresponding results for the tuned images of the real slice. Applications of these estimates in the study of critically non-recurrent real quadratics as well as biaccessible points of quadratic Julia sets are given.

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arXiv1998

Dynamics of Singular Holomorphic Foliations on the Complex Projective Plane

This manuscript is an introduction to the theory of holomorphic foliations on the complex projective plane. Historically the subject has emerged from the theory of ODEs in the complex domain and various attempts to solve Hilbert's 16th Problem, but with the introduction of complex algebraic geometry, foliation theory and dynamical systems, it has now become an interesting subject of its own.

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arXiv1998

Mating Siegel Quadratic Polynomials

Let F be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers theta and nu. Using a new degree 3 Blaschke product model for the dynamics of F and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that F can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers theta and nu.

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arXiv1998

On Dynamics of Cubic Siegel Polynomials

Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number theta, with theta being a given irrational number of Brjuno type. Our main goal is to prove that when theta is of bounded type, the boundary of the Siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial. We also prove that the locus of all cubics with both critical points on the boundary of their Siegel disk is a Jordan curve, which is in some sense parametrized by the angle between the two critical points. A main tool in the bounded type case is a related space of degree 5 Blaschke products which serve as models for our cubics. Along the way, we prove several results about the connectedness locus of these cubic polynomials.

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arXiv1998

Biaccessiblility in quadratic Julia sets II: The Siegel and Cremer cases

Let $f$ be a quadratic polynomial which has an irrationally indifferent fixed point $α$. Let $z$ be a biaccessible point in the Julia set of $f$. Then: 1. In the Siegel case, the orbit of $z$ must eventually hit the critical point of $f$. 2. In the Cremer case, the orbit of $z$ must eventually hit the fixed point $α$. Siegel polynomials with biaccessible critical point certainly exist, but in the Cremer case it is possible that biaccessible points can never exist. As a corollary, we conclude that the set of biaccessible points in the Julia set of a Siegel or Cremer quadratic polynomial has Brolin measure zero.

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