arXiv subjects

Suzanne Boyd

Publications and source records attributed to Suzanne Boyd.

11 recordsLinked to original sources

arXiv2026

Computability of Julia sets for complex H\'enon maps: The role of attracting and neutral cycles

We study the computability of Julia sets for polynomial diffeomorphisms of $\mathbb{C}^2$ with dynamical degree $d>1$, whose prototypical examples are complex H\'enon maps. In previous work, we established computability under the assumption of hyperbolicity (Axiom A). Here, we extend this result to maps whose Fatou components are attracting basins, allowing for the possibility of no attracting basins or infinitely many basins. This yields computability of the Julia set for several classes of non-hyperbolic maps, including certain substantially dissipative maps in the Lyubich-Peters class, and certain quasi-hyperbolic maps. Our proof is based on an algorithm that separates the dynamics into escaping, attracting, and saddle regimes. A key ingredient is the use of stable manifolds of saddle periodic points to approximate the forward Julia set via backward iteration. We first prove the result for generalized H\'enon mappings and then extend it to arbitrary polynomial diffeomorphisms by expressing them as finite compositions of generalized H\'enon maps. Finally, we present examples of non-computability in the presence of neutral dynamics, including H\'enon maps with computable coefficients exhibiting semi-Siegel behavior. These examples show that the computability/non-computability dichotomy associated with neutral dynamics in one-dimensional complex dynamics in part persists in higher dimensions.

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arXiv2026

Paper Fortune Tellers in Julia sets of Generalized McMullen maps II: Sidecars and Zippers

We study the family of complex rational functions known as Generalized McMullen maps, F(z) = z^n + a/z^n+b, for integer n at least 3 fixed, and complex parameters a, b with a nonzero. In prior work by the same authors, we provided a combinatorial model for a large class of maps whose Julia sets contain both infinitely many homeomorphic copies of quadratic Julia sets conjugate to the ``basilica'', and infinitely many subsets homeomorphic to a set which is obtained by starting with the basilica, then changing a finite number of pairs of external ray landing point identifications, following an algorithm we described. In this article, we generalize beyond the basilica, and provide a catalog of additional types of hyperbolic Julia sets of Generalized McMullen maps, where the ``baby'' Julia set can be any rabbit, aeroplane, or Kokopelli quadratic Julia set; that is, where the c-value can be taken from any bulb attached to the main cardioid of the Mandelbrot set, or from the main cardioid of any principal baby Mandelbrot set (no renormalizations).

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arXiv2026

From Ergodic Theory and Probability to Fractal Geometry and Dynamics: Themes in the Work of Manfred Denker

This article surveys the mathematical contributions of Manfred Denker, with a focus on themes that connect ergodic theory, probability theory, dynamical systems, fractal geometry, and statistics. Denker's highly influential work includes a systematic study of the statistical properties of dynamical systems, the development of limit theorems for dependent processes, and the use of thermodynamic formalism to relate geometric and measure-theoretic properties. Particular emphasis is placed on the emergence of probabilistic behavior in deterministic systems, including central limit theorems, invariance principles or local limit theorems, under weak dependence assumptions or in infinite measure. Further topics include equilibrium states and transfer operator methods, the role of conformal measures in fractal geometry, and the asymptotic theory of statistical procedures for dependent data, such as rank statistics and U-statistics. In addition to these theoretical developments, the survey highlights contributions connecting rigorous analysis with computational and statistical methods. Taken together, these works illustrate a unifying perspective in which ergodic, probabilistic, geometric, and statistical methods interact in the study of dynamical systems.

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arXiv2026

Computability properties of hyperbolic complex H\'{e}non maps

In this article, we provide the first theoretical framework guaranteeing that computers can, in principle, be used to analyze the parameter space of complex H\'{e}maps. More precisely, we obtain computability results for hyperbolic polynomial diffeomorphisms of $\mathbb{C}^2$, for which H\'{e}non maps are prototypical examples. Specifically, we establish computability of the Julia set for hyperbolic maps, semi-decidability of hyperbolicity, and lower computability of the hyperbolicity locus in the parameter space of generalized H\'{e}non mappings of fixed degree at least two. Our approach builds upon techniques developed in our's recent previous works on polynomial maps of $\mathbb{C}$ and polynomial skew products of $\mathbb{C}^2$. In the setting of polynomial diffeomorphisms of $\mathbb{C}^2$, however, establishing hyperbolicity for the Julia set is considerably more difficult, as it requires identifying unstable (and stable) cone fields that are preserved and expanded by $Df$ (respectively $Df^{-1}$), and also due to the lack of algorithmically detectable quantitative shadowing.

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arXiv2025

Exploring baby Julia sets in parameter space slices for Generalized McMullen Maps

For the family of complex rational functions of the form R(z)= z^n + a/z^n+b, known as "Generalized McMullen maps", for non-zero a, and integer n fixed and at least 3, we describe the apparent phenomena of baby Julia sets in parameter space appearing both in slices with independent critical orbits and a slice defined by imposing a critical orbit relation. Specifically, we introduce the subfamily where one of two critical orbits is set to be a super-attracting fixed point, provide some general results on this subfamily and describe how Julia set copies in the parameter space slice occur--due to parameters for which the other critical orbit is in the (not immediate) basin of attraction of this fixed critical point. We provide several conjectures on this intriguing phenomena to catalyze further study.

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arXiv2025

A dynamical algorithm to compute hyperbolic Julia sets in polynomial time

Hyperbolic Julia sets of complex polynomials are known to be computable in polynomial time due to pioneering work of Braverman in 2005 (10.1016/j.entcs.2004.06.031). In this paper, we present an alternative method for establishing poly-time computability of hyperbolic Julia sets, which allows us to establish, via a new algorithm, lower computability of the hyperbolicity locus of polynomials of a fixed degree. We first adapt our recently developed algorithms for the computability of polynomial skew products (preprint available arXiv.2508.08033) and then apply a refinement that allows us to establish poly-time computation of hyperbolic Julia sets. Finally, we derive lower computability of the hyperbolicity locus via an adapted lattice/refinement search algorithm. In contrast to Braverman's 2005 algorithm/proof, our approach is dynamical in nature and does not rely on techniques unique to complex analysis.

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arXiv2025

Computability for Axiom A Polynomial Skew Products of $\mathbb{C}^2$

The computability of Julia sets of rational maps on the Riemann sphere has been intensively studied in recent years (see, e.g. https://doi.org/10.17323/1609-4514-2008-8-2-185-231, https://doi.org/10.1090/conm/797/15936) for an overview. For example, by Braverman's results (https://doi.org/10.1016/j.entcs.2004.06.031, https://doi.org/10.1088/0951-7715/19/6/009), hyperbolic and parabolic Julia sets are computable in polynomial time. In this paper, we present the first work on computability related to maps of more than one complex dimension. We examine a family of polynomial endomorphisms of $\mathbb{C}^2$, the polynomial skew products; i.e., maps of the form $f(z,w) = (p(z), q(z,w)),$ where $p$ and $q$ are complex polynomials of the same degree $d\geq 2$. We show that if a polynomial skew product is Axiom A, then its chain recurrent set, which is equal to its non-wandering set and also equal to the closure of the periodic orbits, is computable. Our algorithm also identifies the various hyperbolic sets of different types, i.e., expanding, attracting, and hyperbolic sets of saddle-type. One consequence of our results is that Axiom A is a semi-decidable property on the closure of the Axiom A polynomial skew product locus. Finally, we introduce an algorithm that establishes the lower semi-computability of the hyperbolicity locus of polynomial skew products of a fixed degree.

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arXiv2025

Paper Fortune Tellers in the combinatorial dynamics of some generalized McMullen maps with both critical orbits bounded

For the family of complex rational functions known as "Generalized McMullen maps", F(z) = z^n + a/z^n+b, for complex parameters a and b, with a nonzero, and any integer n at least 3 fixed, we reveal, and provide a combinatorial model for, some new dynamical behavior. In particular, we describe a large class of maps whose Julia sets contain both infinitely many homeomorphic copies of quadratic Julia sets and infinitely many subsets homeomorphic to a set which is obtained by starting with a quadratic Julia set, then changing a finite number of pairs of external ray landing point identifications, following an algorithm we will describe.

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arXiv2024

Baby Mandelbrot sets and Spines in some one-dimensional subspaces of the parameter space for generalized McMullen Maps

For the family of complex rational functions of the form $R_{n,c,a}(z) = z^n + \dfrac{a}{z^n}+c$, known as ``Generalized McMullen maps'', for $a\neq 0$ and $n \geq 3$ fixed, we study the boundedness locus in some one-dimensional slices of the $(a,c)$-parameter space, by fixing a parameter or imposing a relation. First, if we fix $c$ with $|c|\geq 6$ while allowing $a$ to vary, assuming a modest lower bound on $n$ in terms of $|c|$, we establish the location in the $a$-plane of $n$ ``baby" Mandelbrot sets, that is, homeomorphic copies of the original Mandelbrot set. We use polynomial-like maps, introduced by Douady and Hubbard and applied for the subfamily $R_{n,a,0}$ by Devaney. Second, for slices in which $c=ta$, we again observe what look like baby Mandelbrot sets within these slices, and begin the study of this subfamily by establishing a neighborhood containing the boundedness locus.

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arXiv2022

The Boundedness Locus and baby Mandelbrot sets for some generalized McMullen maps

In this paper we study rational functions of the form $ R_{n,a,c}(z) = z^n + \dfrac{a}{z^n} + c, $ with $n$ fixed and at least $3$, and hold either $a$ or $c$ fixed while the other varies. We locate some homeomorphic copies of the Mandelbrot set in the $c$-parameter plane for certain ranges of $a$, as well as in the $a$-plane for some $c$-ranges. We use techniques first introduced by Douady and Hubbard, that were applied for the subfamily $R_{n,a,0}$ by Robert Devaney. These techniques involve polynomial-like maps of degree two.

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arXiv2014

On diffeomorphisms of compact 2-manifolds with all nonwandering points periodic

The aim of the present paper is to study conditions under which all the non-wandering points are periodic points, for a discrete dynamical system of two variables defined on a compact manifold. We include a survey of known results in all dimensions, and study the remaining open question in dimension two. We present two results, one positive and one negative. The negative result: we construct a Kupka--Smale diffeomorphism in $\mathbb{R}^2$ (which can be extended to a diffeomorphism of the sphere) with a closed set of periodic points that differs from the set of nonwandering points. The positive result: we present a condition on the widely studied H\'{e}non family which guarantees that all nonwandering points are periodic. Finally, we close by describing what future work may be needed to resolve our broad goals.

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