The Biquaternion Quadratic Map and Its Julia Sets

Introduction

In the complex plane the quadratic family is one map, $\zeta\mapsto\zeta^2+C$, and one filled Julia set for each parameter. In the biquaternion algebra $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ the same formula denotes several maps at once, because the square of an element can be read with the general plain bilinear product, with one of the three other products the space carries, or with the left, the right and the matrix product, and the maps so obtained are not equivalent. This article fixes the reading, defines the family and its filled Julia set, and separates what survives the passage from the field to the algebra from what does not.

The algebra, its conjugations and its remarkable subspaces are The Biquaternion Algebra and Its Vector Spaces and Introduction to the Remarkable Subspaces; its zero divisors and its idempotents are Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, §The Zero Divisors and Square Roots of One and the Idempotents of the Biquaternion Algebra; the norm and the Euclidean norm are Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra; the matrix model is Introduction to the 2×2 Matrix Element Representation $M_2(\mathbb{C})$ of Biquaternions. These are used here and none is restated. The classical theory in $\mathbb{C}$ is The Julia Sets of a Complex Polynomial, The Mandelbrot Set and the Quadratic Family and The Fatou Components and the Classification of the Dynamics, and the comparison with it is made section by section.

This article owns the several readings of the square and the choice of the map, the definition of the biquaternion quadratic family, the filled Julia set and the Julia set, their complete invariance, the equivariance of the family under the conjugations, the critical set, and the reduction of the central-parameter family to a pair of complex quadratic families in the matrix model. It does not treat the escape radius (The Escape Radius and the Green's Function for the Biquaternions), the parameter locus (The Biquaternion Mandelbrot Set and the Connectedness Locus), the slices (The Slices of the Biquaternion Julia Sets), the zero divisors as a dynamical phenomenon (The Zero Divisors and the Singular Julia Sets), the Jacobian and the Fatou theory (The Biquaternion Holomorphic Dynamics and the Jacobian), or the dimension (The Hausdorff Dimension of the Biquaternion Julia Sets). No physical vocabulary is used.

Standing convention. Throughout, $\tilde Q=\sum_\mu Q_\mu e_\mu$ with $Q_\mu\in\mathbb{C}$ is a biquaternion, $\tilde Q_5$ is the parameter, and $\tilde Q^2$ is the square in the general plain bilinear product, so that the family is

$$ F_{\tilde Q_5}(\tilde Q)=\tilde Q^2+\tilde Q_5, $$

the sum being that of the underlying complex vector space. The parameter is fixed and the article writes $F$ for $F_{\tilde Q_5}$ where no confusion results.

The Several Interpretations of the Square

The formula $\tilde Q\mapsto\tilde Q^2$ is unambiguous once a product is chosen and the square is read as the diagonal of the corresponding bilinear map. Both clauses matter.

The general plain bilinear square. The general plain bilinear product $\tilde Q\tilde Q_2$ is the only one of the four general products of the space that is the multiplication of an associative unital $\mathbb{C}$-algebra (The Biquaternion Algebra and Its Vector Spaces). Its diagonal $\tilde Q\tilde Q$ is the square this article uses. In the four complex coordinates it is a polynomial map with no conjugated coordinate, hence holomorphic; this is what makes the matrix model above all a model of complex dynamics.

The diagonal of a left or a right multiplication. For a fixed element $\tilde Q_3$ the two maps

$$ \tilde Q_1\mapsto \tilde Q_3\tilde Q_1+\tilde Q_5\quad (\text{left}), \qquad \tilde Q_1\mapsto \tilde Q_1\tilde Q_3+\tilde Q_5\quad (\text{right}) $$

are affine: for $\tilde Q_3$ a unit they are conjugate to a linear map, their orbits are exponentials, and their Julia set is empty. They return the quadratic family only on the diagonal $\tilde Q_1=\tilde Q=\tilde Q_3$. The left and the right products give the square only on the diagonal; off the diagonal they give two affine families with no fractal content. The matrix product agrees with the general plain bilinear square on the nose, because $\mathsf{M}_2$ is an algebra isomorphism (Introduction to the 2×2 Matrix Element Representation $M_2(\mathbb{C})$ of Biquaternions).

The conjugate squares. Replacing one factor by a conjugate gives a map of a different kind. With the natural conjugate, $\tilde Q\tilde Q^{\natural}=\langle\tilde Q\rangle_{\natural}e_t$ is central and scalar, so the "natural quadratic family" is $\tilde Q\mapsto \langle\tilde Q\rangle_{\natural}e_t+\tilde Q_5$, a scalar-valued map whose orbit is governed by the complex numbers $\langle F^n(\tilde Q)\rangle_{\natural}$ alone. With the Hermitian conjugate, $\tilde Q\tilde Q^{*}$ is the Hermitian biquaternion whose scalar part is the square of the Euclidean norm, and the "Hermitian quadratic family" is a real-analytic, non-holomorphic map with values in the Hermitian subspace $\mathbb{M}_+$. These are different theories; neither is the theory of this article.

Remark (why the choice is a choice). The general plain bilinear product is the canonical multiplication of $\mathbb{B}$ once $\mathbb{B}$ is presented as a $\mathbb{C}$-algebra with the quaternion relations, so the first line of the table is the canonical reading. What is not canonical is the size. The algebra carries a complex-valued multiplicative norm $\langle\cdot\rangle_{\natural}$, its modulus $r=\sqrt{|N|}$, and the Euclidean norm $\|\cdot\|_E$, and no one of the three is singled out by the algebra. The map is canonical; the norm used to define boundedness is a choice of metric, and the escape radius depends on it. Boundedness itself does not, by the next section.

The Quadratic Family

Definition. The biquaternion quadratic family is the family of self-maps of $\mathbb{B}$

$$ F_{\tilde Q_5}:\mathbb{B}\to\mathbb{B}, \qquad F_{\tilde Q_5}(\tilde Q)=\tilde Q^2+\tilde Q_5, $$

one map for each parameter $\tilde Q_5\in\mathbb{B}$. The orbit of a point $\tilde Q$ is the sequence $\tilde Q_0=\tilde Q$, $\tilde Q_{n+1}=F_{\tilde Q_5}(\tilde Q_n)$.

Proposition (the orbit is a polynomial in the initial point). For every $n$ there is a polynomial $p_n$ of degree $2^n$ with

$$ \boxed{\;p_0(\tilde Q)=\tilde Q, \qquad p_{n+1}(\tilde Q)=p_n(\tilde Q)^2+\tilde Q_5, \qquad \tilde Q_n=p_n(\tilde Q)\;}. $$

If $\tilde Q_5$ is central, $\tilde Q_5=Ce_t$ with $C\in\mathbb{C}$, every $p_n$ has coefficients in the centre and $\tilde Q_n$ is a polynomial in $\tilde Q$ alone.

Proof. Induction on $n$: $\tilde Q_{n+1}=p_n(\tilde Q)^2+\tilde Q_5=p_{n+1}(\tilde Q)$ by the recursion. If $\tilde Q_5$ is central the recursion stays in the polynomial ring over the centre, and the evaluation is taken in the commutative subalgebra $\mathbb{C}[\tilde Q]$ generated by one element.

The last clause is why the central parameter is the computable one: $\mathbb{C}[\tilde Q]$ has complex dimension at most two, and a polynomial in one element is a polynomial in its two coordinates there (The Biquaternion Algebra and Its Vector Spaces, §The Subalgebra Generated by One Element).

The Filled Julia Set and the Julia Set

Definition. The filled Julia set of $F_{\tilde Q_5}$ is the set of points whose orbit is bounded,

$$ \mathcal K_{\tilde Q_5}=\{\tilde Q\in\mathbb{B} : (F_{\tilde Q_5}^{n}(\tilde Q))_{n\ge0} \text{ is bounded}\}, $$

and the Julia set is its boundary, $J_{\tilde Q_5}=\partial\mathcal K_{\tilde Q_5}$.

Remark (boundedness does not depend on the norm). In the finite-dimensional real vector space $\mathbb{B}\cong\mathbb{R}^8$ all norms are equivalent, so a sequence is bounded for one norm exactly when it is bounded for every norm. The filled Julia set is a set-theoretic object of the algebra and not of a chosen metric. This is the sharpest contrast with the escape radius, which is a metric statement and does depend on the norm (The Escape Radius and the Green's Function for the Biquaternions).

Proposition (complete invariance). $F_{\tilde Q_5}(\mathcal K_{\tilde Q_5})\subset\mathcal K_{\tilde Q_5}$ and $F_{\tilde Q_5}^{-1}(\mathcal K_{\tilde Q_5})=\mathcal K_{\tilde Q_5}$; the same holds with $J_{\tilde Q_5}$ in place of $\mathcal K_{\tilde Q_5}$. Both sets are closed.

Proof. If $\tilde Q\in\mathcal K$ then the orbit of $F(\tilde Q)$ is the orbit of $\tilde Q$ with its first term deleted, hence bounded, so $F(\tilde Q)\in\mathcal K$ and $F(\mathcal K)\subset\mathcal K$. Conversely if $F(\tilde Q)\in\mathcal K$ then the orbit of $\tilde Q$ is the bounded orbit of $F(\tilde Q)$ preceded by the single point $\tilde Q$, hence bounded, so $\tilde Q\in\mathcal K$; thus $F^{-1}(\mathcal K)=\mathcal K$. For the Julia set: if $\tilde Q\in J=\partial\mathcal K$ and $V$ is a neighbourhood of $F(\tilde Q)$, then $F^{-1}(V)$ is a neighbourhood of $\tilde Q$, so it meets $\mathcal K$ — giving $V\cap\mathcal K\neq\emptyset$ because $F(\mathcal K)\subset\mathcal K$ — and it meets the complement of $\mathcal K$ — giving a point whose image lies in $V$ and, since $F^{-1}(\mathcal K)=\mathcal K$, outside $\mathcal K$. Hence $F(J)\subset J$. Conversely if $F(\tilde Q)\in J$ then $\tilde Q\in F^{-1}(\mathcal K)=\mathcal K$; if $\tilde Q$ were interior to $\mathcal K$ then $F(\tilde Q)$ would be interior, because a non-constant holomorphic map is open, contradicting $F(\tilde Q)\in\partial\mathcal K$; hence $\tilde Q\in J$ and $F^{-1}(J)\subset J$; the inclusion $J\subset F^{-1}(J)$ is the previous one. Closedness of $\mathcal K$ is that of a sublevel set of a continuous function, and of $J$ that of a boundary.

No properness of $F$ was used, and none holds: the equation $\tilde Q^2=\tilde Q_4$ has the whole cone of square-zero elements as its fibre over $\tilde Q_4=0$.

The Conjugations and the Equivariance

Theorem (equivariance). Let $g$ be one of the three conjugations ${}^{\natural}$, $\bar{\cdot}$, ${}^{*}$ — the non-identity elements of the group of involutions — and let $g$ act coefficientwise on the parameter. Then

$$ \boxed{\;g\circ F_{\tilde Q_5}=F_{g(\tilde Q_5)}\circ g\;}. $$

Consequently $g(\mathcal K_{\tilde Q_5})=\mathcal K_{g(\tilde Q_5)}$ and $g(J_{\tilde Q_5})=J_{g(\tilde Q_5)}$.

Proof. Each of the three maps is an automorphism or an anti-automorphism of the ring, so $g(\tilde Q^2)=(g\tilde Q)^2$; and each is real-linear or conjugate-linear, hence preserves boundedness of a sequence. Applying $g$ to $F_{\tilde Q_5}(\tilde Q)=\tilde Q^2+\tilde Q_5$ gives $(g\tilde Q)^2+g\tilde Q_5=F_{g(\tilde Q_5)}(g\tilde Q)$.

Remark (the reversal is excluded by a sign). The fourth conjugation, the reversal $\flat$, satisfies $(\tilde Q\tilde Q_2)^{\flat}=-\tilde Q_2^{\flat}\tilde Q^{\flat}$, so $(\tilde Q^2)^{\flat}=-(\tilde Q^{\flat})^2$ and $\flat\circ F_{\tilde Q_5}=F^{-}_{\flat(\tilde Q_5)}\circ\flat$, where $F^{-}_{\tilde Q_5'}(\tilde Q)=-\tilde Q^2+\tilde Q_5'$ is the odd quadratic family. The reversal carries the even family to the odd family and is not a symmetry of the even fractal (The Slices of the Biquaternion Julia Sets, §The Conjugation Symmetries of the Slices).

Corollary (the symmetry group of the fractal). Let $G_{\tilde Q_5}=\{g\in\{\mathrm{id},{}^{\natural},\bar{\cdot},{}^{*}\} : g(\tilde Q_5)=\tilde Q_5\}$. Then $\mathcal K_{\tilde Q_5}$ and $J_{\tilde Q_5}$ are invariant under every element of $G_{\tilde Q_5}$. The stabiliser is one of the five subgroups of the Klein group, according as the parameter is a real central element, a non-real central element, a non-central real quaternion, a non-central Hermitian element, or a general element; the classification, the five symmetric families and the central inversion are The Slices of the Biquaternion Julia Sets, §The Conjugation Symmetries of the Slices.

Proof. The invariance is the theorem applied elementwise; the classification is the fixed-subspace computation of The Group of Involutions and Introduction to the Remarkable Subspaces, carried out in the cited section.

The Critical Set

Definition. The critical set of $F_{\tilde Q_5}$ is the set of points where the complex derivative $dF_{\tilde Q_5}$ fails to be invertible.

Proposition (the critical set). For every parameter, the derivative of $F_{\tilde Q_5}$ at $\tilde Q$ is the sum of the left and the right multiplication,

$$ dF_{\tilde Q_5}|_{\tilde Q}(\tilde Q_2)=\tilde Q\tilde Q_2+\tilde Q_2\tilde Q , $$

and its determinant is, up to a fixed non-zero factor depending only on the identification of $\mathbb{B}$ with $\mathbb{C}^4$,

$$ \boxed{\;\det dF_{\tilde Q_5}|_{\tilde Q} \;=\; 4\,\langle\tilde Q\rangle_{\natural}\,(2Q_t)^2\;}. $$

Hence the critical set is the union of the zero-divisor cone $\{\langle\tilde Q\rangle_{\natural}=0\}$ and the vector subspace $\{Q_t=0\}=\mathrm{Vect}(\mathbb{B})$.

Proof. The family is $\tilde Q^2$ plus a constant, so its derivative is that of the general plain bilinear square, which is $\tilde Q_2\mapsto\tilde Q\tilde Q_2+\tilde Q_2\tilde Q$ by the product rule. Transport along the algebra isomorphism $\mathsf{M}_2$ to $M_2(\mathbb{C})$: in the coordinates $(M_{11},M_{12},M_{21},M_{22})$ the linear map $V\mapsto MV+VM$ has matrix $I\otimes M+M^{\mathsf{T}}\otimes I$, whose determinant is $\det(M)\operatorname{tr}(M)^2\cdot 4$; substituting $\det\mathsf{M}_2(\tilde Q)=\langle\tilde Q\rangle_{\natural}$ and $\operatorname{tr}\mathsf{M}_2(\tilde Q)=2Q_t$ gives the formula. The determinant is a polynomial in $\tilde Q$ that is not identically zero, so its zero set is a hypersurface of the complex four-space; by the product formula that zero set is the union of the quadric $\langle\tilde Q\rangle_{\natural}=0$ and the hyperplane $Q_t=0$, which is the statement.

Counting the coordinates, the critical set is the zero set of that one non-zero polynomial, hence a hypersurface of the complex four-space, with the two components named. The zero-divisor cone is the quadric $\langle\tilde Q\rangle_{\natural}=0$ and the vector subspace is the hyperplane $Q_t=0$.

Remark. The critical set is larger than the zero-divisor cone: it also contains the whole vector subspace $\mathrm{Vect}(\mathbb{B})$ of pure biquaternions, where the derivative degenerates because $Q_t=0$ makes the two eigenvalues of $\mathsf{M}_2(\tilde Q)$ opposite. The derivative is singular exactly on the union of the zero divisors and the pure vectors, a hypersurface with two components. The singular set governs the Fatou theory (The Biquaternion Holomorphic Dynamics and the Jacobian).

The Central Parameter and the Eigenvalue Reduction

Theorem (central parameter, eigenvalue reduction). Let $\tilde Q_5=Ce_t$ with $C\in\mathbb{C}$, let $q_n$ be the iterates of the complex quadratic map $\zeta\mapsto\zeta^2+C$ starting from $\zeta_0=\zeta$, and let $p_n$ be as above. For every $\tilde Q$,

$$ \mathsf{M}_2(F_{\tilde Q_5}^{n}(\tilde Q))=p_n\bigl(\mathsf{M}_2(\tilde Q)\bigr)=q_n\text{ applied to the spectrum}, $$

in the sense that the spectrum of $\mathsf{M}_2(F_{\tilde Q_5}^{n}(\tilde Q))$ is $\{q_n(\lambda_1),q_n(\lambda_2)\}$ for the spectrum $\{\lambda_1,\lambda_2\}$ of $\mathsf{M}_2(\tilde Q)$. If $\mathsf{M}_2(\tilde Q)$ is diagonalisable, then $\tilde Q\in\mathcal K_{\tilde Q_5}$ if and only if both eigenvalues lie in the filled Julia set $K_C$ of $\zeta\mapsto\zeta^2+C$. If $\mathsf{M}_2(\tilde Q)$ has a repeated eigenvalue $\lambda$ and is not scalar, the extra condition is that the derivative sequence $(q_n'(\lambda))$ be bounded as well.

Proof. For a polynomial $p$ and a matrix $M$, the eigenvalues of $p(M)$ are the values of $p$ on the eigenvalues of $M$; this is the statement that the spectrum is transported by polynomial functional calculus, and it holds for every polynomial without any hypothesis on $M$. Boundedness: if $M=SJS^{-1}$ with $J$ the Jordan form, then $p_n(M)=Sp_n(J)S^{-1}$, and $p_n(J)$ is diagonal with entries $p_n(\lambda_j)$ when the eigenvalues are distinct, bounded exactly when each $(p_n(\lambda_j))$ is bounded, and is a Jordan block $\left(\begin{smallmatrix}p_n(\lambda)&p_n'(\lambda)\\0&p_n(\lambda)\end{smallmatrix}\right)$ for a repeated non-scalar eigenvalue, bounded exactly when both $(p_n(\lambda))$ and $(p_n'(\lambda))$ are. Norm equivalence finishes the boundedness claim.

Corollary ($\tilde Q_5=0$). For the map $F_0(\tilde Q)=\tilde Q^2$ the iterated polynomial is $p_n(x)=x^{2^n}$, so the spectrum of $\mathsf{M}_2(F_0^n(\tilde Q))$ is $\{\lambda_1^{2^n},\lambda_2^{2^n}\}$ and

$$ \mathcal K_0=\{\tilde Q : \rho(\mathsf{M}_2(\tilde Q))<1\}\cup\{\tilde Q : \rho(\mathsf{M}_2(\tilde Q))=1 \text{ and } \mathsf{M}_2(\tilde Q) \text{ diagonalisable}\} . $$

In particular every square-zero element has $\rho(\mathsf{M}_2(\tilde Q))=0$ and so lies in $\mathcal K_0$, however large its Euclidean norm is.

Proof. $q_n(\zeta)=\zeta^{2^n}$ and $q_n'(\zeta)=2^n\zeta^{2^n-1}$. For $|\lambda|<1$ the orbit tends to $0$; for $|\lambda|>1$ it escapes; for $|\lambda|=1$ the orbit stays on the unit circle and is bounded, while $|q_n'(\lambda)|=2^n$ is unbounded, so a non-scalar Jordan block at $|\lambda|=1$ makes the iterate unbounded and a diagonalisable one does not. A square-zero element is a non-zero nilpotent $2\times2$ matrix, of spectrum $\{0,0\}$ and not a Jordan block at a non-zero eigenvalue, so its iterate is $0$ from the first step and lies in $\mathcal K_0$.

Example (the square-zero element). The element $\tilde Q=\vec e_x+i\vec e_y$ is a pure vector with $(\vec Q,\vec Q)=1+i^2=0$, hence $\tilde Q^2=0$; its orbit under $F_0$ is $\tilde Q,0,0,\dots$, so it lies in $\mathcal K_0$ at distance $\sqrt2$ from the origin. It is the smallest witness that the filled Julia set is not confined to a bounded ball and that no escape radius can be stated in the Euclidean norm: an element of norm arbitrarily large and square-zero maps to $0$ in one step.

Summary

The biquaternion quadratic family is $F_{\tilde Q_5}(\tilde Q)=\tilde Q^2+\tilde Q_5$ in the general plain bilinear product, the only one of the four general products that makes the space an associative unital $\mathbb{C}$-algebra; the other readings of the square give a scalar map for the natural conjugate, a real-analytic map for the Hermitian conjugate, and two affine families for the left and right multiplications, the last coinciding with the quadratic family only on the diagonal. The filled Julia set $\mathcal K_{\tilde Q_5}$ is the set of bounded orbits and the Julia set is its boundary; both are closed and completely invariant, and boundedness is independent of the norm because the algebra is finite-dimensional. The family is equivariant under the three conjugations of the group of involutions, $g\circ F_{\tilde Q_5}=F_{g(\tilde Q_5)}\circ g$, the reversal being excluded because it carries the even family to the odd one; so the fractal carries the stabiliser of the parameter, which is the whole group for a real central parameter, the subgroup $\{\mathrm{id},{}^{\natural}\}$ for a non-real central parameter, $\{\mathrm{id},\bar{\cdot}\}$ for a non-central real quaternion, $\{\mathrm{id},{}^{*}\}$ for a non-central Hermitian element, and the trivial group otherwise. The derivative is $\tilde Q_2\mapsto\tilde Q\tilde Q_2+\tilde Q_2\tilde Q$ and its determinant is proportional to $\langle\tilde Q\rangle_{\natural}Q_t^2$, so the critical set is the union of the zero-divisor cone and the vector subspace of pure vectors. For a central parameter the whole dynamics reduces, through the matrix model, to the complex quadratic map acting on the two eigenvalues; the reduction is exact for a diagonalisable matrix and acquires a derivative condition on a non-scalar Jordan block, and for the parameter zero it gives $\mathcal K_0$ as the set of matrices with spectral radius below one together with the diagonalisable matrices of spectral radius one.

Summary of Notation

Further Reading

• The Biquaternion Algebra and Its Vector Spaces, for the algebra, its one associative unital product and the plane generated by one element.
• Introduction to the 2×2 Matrix Element Representation $M_2(\mathbb{C})$ of Biquaternions, for the isomorphism $\mathsf{M}_2$, the trace and the determinant.
• The Group of Involutions, for the four conjugations, their fixed spaces and the orbits that give the symmetry groups.
• The Slices of the Biquaternion Julia Sets, for the five symmetric families and the slices on which the symmetries are read.
• The Julia Sets of a Complex Polynomial, The Mandelbrot Set and the Quadratic Family and The Fatou Components and the Classification of the Dynamics, for the one-variable theory the family generalises.
• The Escape Radius and the Green's Function for the Biquaternions, for the metric statement the filled Julia set dispenses with.