Complex dynamics

Complex dynamics studies iteration of holomorphic and rational maps, especially the division of the Riemann sphere into stable Fatou behavior and chaotic Julia behavior.

Holomorphic maps $R$ and $S$ are holomorphically conjugate if a biholomorphic coordinate change $h$ satisfies $h\circ R=S\circ h$. Iterating gives $h\circ R^n=S^n\circ h$, so periodic points and their exact periods correspond. Differentiating at a periodic point gives equality of return multipliers, since the nonzero derivative of $h$ cancels. Locally uniform convergence and normality transfer through the coordinate change, so it also transports Fatou sets and Julia sets. Between entire Riemann spheres the biholomorphic change is a Möbius transformation; an affine change is sufficient for many polynomial normalizations.

Two rational sphere maps are Möbius conjugate if $S=h\circ R\circ h^{-1}$ for a Möbius transformation $h$. This is precisely global holomorphic conjugacy between Riemann-sphere self-maps. It preserves degree, local degrees of a holomorphic map, exact periods and return periodic-orbit multipliers, and transports the Fatou set and Julia set by $h$. Since a Möbius map and its inverse are Lipschitz on the compact sphere, this last assertion follows directly from equicontinuity of the iterate families. An affine holomorphic conjugacy is the special case where $h$ fixes infinity.

A subset $E$ is completely invariant for a rational map if $R(E)=E=R^{-1}(E)$. This asserts invariance in both directions, including every preimage, rather than just invariance of one chosen inverse branch. For a nonconstant sphere map, the two inclusions $R(E)\subseteq E$ and $R^{-1}(E)\subseteq E$ already imply the equalities, because the map is surjective. The Fatou set and Julia set are completely invariant. A Fatou component need not itself be completely invariant even though the union of all components is.

A point $p$ is periodic for an iterated map $R$ if $R^n(p)=p$ for some positive integer $n$. Its exact period is the least such $n$, and the distinct points $p,R(p),\dots ,R^{n-1}(p)$ form its periodic orbit. For a holomorphic map its periodic-orbit multiplier is $(R^n{)}^{\prime }(p)$, computed in a local coordinate; at infinity use a reciprocal coordinate. The chain rule makes this the cyclic product of derivatives along the orbit, so it is the same at every point of that orbit.

An exact $n$-cycle consists of $n$ distinct points $w_0,\dots ,w_{n-1}$ with $R(w_j)=w_{j+1}$, indices read modulo $n$. Every point has exact period $n$. The return multiplier is ${\prod }_{j=0}^{n-1}{R}^{\prime }(w_j)$, computed in local charts if necessary; cyclically changing the starting point leaves it unchanged. Thus an attracting, superattracting or repelling return is of the same type at every point of the cycle. If a period-$n$ root coincides with a lower-period point at a parameter limit, multiplicity in the equation $R^n(z)=z$ need not represent distinct points of an exact cycle.

A holomorphic periodic point is repelling when its return multiplier has modulus greater than one. It belongs to the Julia set: if the iterates were normal near it, a subsequence of the return iterates, all fixing the point, would converge locally uniformly in the spherical metric. Near their fixed finite value the limit lies in a finite coordinate chart, so Cauchy's derivative estimate would bound their derivatives at the point. Those derivatives are the successive powers of the multiplier and are unbounded. The same argument in a reciprocal coordinate handles infinity.

A holomorphic periodic point is attracting when its return multiplier has modulus strictly less than one. For the return map $G=R^n$, choose a local coordinate at the fixed point and a number $q$ strictly between $|G^{\prime }(p)|$ and one. Taylor expansion gives $|G(z)-p|\le q|z-p|$ on a sufficiently small disc. It maps that disc into itself, and successive returns converge uniformly to $p$ there. Composing with the finitely many intervening maps gives locally uniform convergence to the periodic orbit and places its basin in the Fatou set.

A holomorphic periodic point is superattracting if its return multiplier is zero. The return map then has local expansion $G(p+u)=p+a{u}^e+O(u^{e+1})$ with $e\ge 2$. This implies a strict contraction on a sufficiently small disc and rapid convergence of returns. Infinity is superattracting for every polynomial of degree at least two: the reciprocal-coordinate return has a zero of order equal to the polynomial degree.

For $f_c(z)=z^2+c$, the Mandelbrot set is $\mathcal{M}=\{c:\{f_c^n(0){\}}_{n\ge 0}\text{ is bounded}\}$. Thus it records the nonescaping finite critical orbit. The connected Julia set criterion for a polynomial says that the corresponding filled Julia set is connected exactly for these quadratic parameters. Parameters with an attracting fixed point form the main cardioid: its multiplier $\lambda$ gives the fixed point $\lambda /2$ and parameter $c=\lambda /2-{\lambda }^2/4$, with $|\lambda |<1$. Its boundary has a cusp at $c=1/4$. There the critical orbit increases along the real interval to the parabolic fixed point $1/2$, so the cusp belongs to the set. The real section extends from $-2$ to $1/4$; the familiar period-two bulb is attached at $-3/4$.

The period-two bulb is the open disc $|c+1|<1/4$ of quadratic parameters with an attracting exact two-cycle. The points in that cycle are roots of $z^2+z+c+1$, whose product is $c+1$. Its multiplier is therefore $4(c+1)$, giving precisely the disc inequality. At $c=-3/4$ the two roots merge into the fixed point $-1/2$, whose original multiplier is minus one; the second iterate has multiplier one and a triple fixed-point root. This boundary point is the attachment to the main cardioid.

The main cardioid consists of quadratic parameters with an attracting fixed point. If that fixed point has multiplier $\lambda$, then it is $\lambda /2$ and $c=\lambda /2-{\lambda }^2/4$, $|\lambda |<1$. The map is injective on the open unit disc: equality of two parameter values forces either equal multipliers or their sum to be two, impossible for distinct interior multipliers. Its boundary is the cardioid obtained from $\lambda =e^{it}$. At $\lambda =1$ the parameter derivative vanishes, giving the cusp $c=1/4$. Parameters just to the real right of that cusp have monotonically escaping critical orbits, while the cusp's critical orbit tends to $1/2$, so it is on the Mandelbrot set boundary.

A family of holomorphic or meromorphic maps is normal when every sequence has a subsequence converging locally uniformly in the spherical metric, possibly to infinity.

For a holomorphic normal family, limits are holomorphic or identically infinity. A uniform bound at one interior point excludes infinity on its connected domain. Every sequence therefore has a finite locally uniform subsequential limit. The Cauchy integral formula makes the derivatives converge locally uniformly too, so the derivative family is normal. Without the point bound this can fail: $n(1+z^2/2)$ tends uniformly to infinity on the unit disk, whereas its derivatives $nz$ are not normal at zero.

Choose a countable cover by connected disks on which a family is normal. From any sequence extract successive subsequences converging on each disk and use the diagonal sequence. The limits agree on overlaps; a finite holomorphic limit and infinity cannot agree on a nonempty overlap. Connectedness therefore gives one global type of limit, and finite compact subcovers give locally uniform convergence. This proves normality of the family on the whole domain from normality near every point.

If a bounded holomorphic function $g$ on the complex upper half-plane has $g(iy)\to \ell$ as $y\downarrow 0$, the rescaled functions $g(z/n)$ form a normal family. Every subsequential limit equals $\ell$ on the imaginary axis, hence everywhere by the identity theorem for holomorphic functions. The entire sequence converges locally uniformly to $\ell$. For a nontangential sequence $z_j=x_j+i{y}_j\to 0$, put $n_j=\lfloor 1/y_j\rfloor$; then $n_j{z}_j$ remains in a fixed compact subset of the half-plane. This gives $g(z_j)\to \ell$. The bounded example $e^{-i/z}$ shows that unrestricted boundary convergence can fail.

// Destination: analysis.bigb

A family of meromorphic functions on a domain that omits three fixed points of the Riemann sphere is normal. For holomorphic functions with values in the complex plane, omitting two fixed values suffices.

The Fatou set of a rational map is the largest open subset of the Riemann sphere on which its iterates form a normal family. Its complement is the Julia set.

A Fatou component is a connected component of the Fatou set of a rational map. It is open. A rational map sends each Fatou component onto another one: complete invariance of the Fatou set puts it inside the inverse image of the target component; its maximal connectedness makes it one full inverse-image component. Restricting the globally proper sphere map to such a component is proper, since the inverse image of a compact subset intersected with the component is closed in a compact set. Its image is both open and closed in the connected target component, hence the whole target. A component is periodic if some iterate returns it to itself.

A Herman ring is a doubly connected periodic Fatou component on which the first-return map is holomorphically conjugate to an irrational rotation of an annulus $r<|w|<1$. Compactness of the circle of rotation factors proves normality of the return iterates, and the finitely many intervening iterates preserve normality. It is the annular rotation-domain counterpart of a Siegel disc.

A Siegel disc is a simply connected periodic Fatou component on which the first-return map is holomorphically conjugate to an irrational rotation of the unit disc: $h\circ R^p\circ h^{-1}(w)=e^{2\pi i\theta }w$, with $\theta \notin \mathbb{Q}$. Its return iterates are normal because any sequence of rotation factors has a convergent subsequence in the unit circle, giving locally uniform limits after conjugation. The iterates generally do not converge to a constant. This is a rotation domain, illustrating stability without attraction.

Suppose a Fatou component $U$ satisfies $R(U)\subseteq U$, and some $a\in U$ has $R^{-1}(a)\subseteq U$. Every connected component of $R^{-1}(U)$ maps properly and surjectively to $U$: properness follows by intersecting compact preimages with the relatively closed component; openness and closedness then give surjectivity. Such a component must meet the entire fiber over $a$, which lies in $U$. Since $U$ is itself one inverse-image component, there can be no other. Hence $R^{-1}(U)=U$ and $R(U)=U$, proving complete invariance.

A periodic Fatou component of a rational map is an immediate attracting basin, an immediate parabolic basin, a Siegel disc, or a Herman ring. Attracting and parabolic basins capture critical orbits, while the boundaries of rotation domains lie in the closure of the postcritical set.

Every immediate basin of an attracting fixed point of a rational map contains a critical point. Otherwise the map on the basin would be an unbranched covering; lifting it to the unit disc would give an automorphism fixing a point, contradicting strict contraction at the attracting fixed point.

The postcritical set is the closure of the union of the forward orbits of all critical values.

A parabolic basin consists of Fatou components whose iterates approach a parabolic periodic orbit.

A parabolic basin consists of points whose iterates converge to a parabolic cycle along a specified attracting direction. Each connected basin component lies in the Fatou set, and its boundary lies in the Julia set.

For $P(z)=z-z^2$, the disc $\Delta =\{|z-1/2|<1/2\}$ corresponds under $w=1/z$ to $\Re w>1$. The transformed map is $T(w)=w+1+1/(w-1)$, so $\Re T(w)>\Re w+1$. Thus $P(\Delta )\subseteq \Delta$ and $|P^n(z)|<1/(n+1)$ there. The Fatou component $U$ containing the disc has $P^n\to 0$ locally uniformly by normality and the identity theorem. It is forward invariant. The point $1/4$ in the disc has the entire fiber $\{1/2\}$, so the Fatou component containing an entire fiber is completely invariant criterion gives $P^{-1}(U)=U$. Zero itself is a Julia point: a normal subsequential limit near zero would be zero by the disc limit, but every iterate has derivative one at zero, contradicting derivative convergence.

Every Fatou component of a rational map of degree at least two is eventually periodic. Thus the periodic-component classification accounts for the entire Fatou set.

The Julia set is the complement of the Fatou set. It is nonempty, closed, completely invariant, and perfect for every rational map of degree at least two.

Let $R,S$ commute. On an open set where $\{R^n\}$ is equicontinuous, so is $\{S\circ R^n\}$, by the rational map is Lipschitz in the chordal metric bound. Since this equals $\{R^n\circ S\}$, equicontinuity descends through the open map $S$: near $S(x)$ every nearby value has a preimage near $x$, and the corresponding estimates hold for all $n$. Thus $S(F(R))\subseteq F(R)$. A degree-at-least-two Julia set contains at least three points, all omitted by the iterates of $S$ on $F(R)$. The three-value Montel theorem gives $F(R)\subseteq F(S)$. Exchanging the maps proves equality of their Fatou and Julia sets. The degree hypothesis excludes Möbius examples with only one Julia point.

For a polynomial $P$ of degree at least two, the filled Julia set $K(P)$ consists of the points with bounded forward orbit. Choose an escape radius $B$ so that $|P(z)|>2|z|$ for $|z|>B$. Then $K(P)={\bigcap }_{n\ge 0}(P^n{)}^{-1}(\{|z|\le B\})$ is compact, and its complement is the basin of infinity. On the interior all iterates are bounded by $B$, so they form a normal family; on the complement they converge locally uniformly to infinity. At a boundary point, normality would give a subsequential limit equal to infinity on an open escaping part and hence identically infinity, contradicting the uniform bound at a nearby point of $K(P)$. Therefore $\partial K(P)=J(P)$.

If a closed set $E$ satisfies $f^{-1}(E)=E$ for a rational map of degree at least two, then either $E$ has at most two points and lies in the Fatou set, or $J(f)\subseteq E$. The complement omits every point of $E$, so Montel theorem proves normality when $|E|\ge 3$.

The accumulation-point set of $J(f)$ is closed and completely invariant. Applying the classification of completely invariant closed sets shows that it equals $J(f)$; hence the Julia set is perfect.

For every $d\ge 2$, choose a $d$th root of unity $r\ne 1$ with $d|1-r|>1$, put $a=1-r$, and setIts only critical points are $0$ and infinity, and both land on the repelling fixed point $r$. The postcritically finite map has no possible Fatou component, so its Julia set is the whole sphere.

The degree-$d$ Chebyshev polynomial has Julia set the interval $[-1,1]$ after the standard normalization. Its Fatou set is the connected complement of that interval in the Riemann sphere, so it has exactly one Fatou component.

Böttcher's equation asks for a holomorphic coordinate conjugating a map with a superattracting fixed point to a monomial. If $f(z)=a{z}^m+O(z^{m+1})$ with $m\ge 2$ and $a\ne 0$, a normalized local solution has $\varphi (z)=bz+O(z^2)$ with $b^{m-1}=a$. The Böttcher theorem asserts the existence and uniqueness of this local coordinate after choosing $b$.

Near a superattracting fixed point, a holomorphic map is conformally conjugate to its leading monomial.

For a degree-$d$ polynomial $f(z)=a_d{z}^d+\cdots$, the Böttcher coordinate near infinity satisfiesIts modulus extends naturally throughout the basin of infinity.

The escape-rate Green function isIt vanishes on the filled Julia set, is positive and harmonic on the basin of infinity, satisfies $G_f\circ f=d{G}_f$, and equals $\log |{\varphi }_f|$ near infinity and by continuation throughout the basin.

The Julia set of a polynomial is connected exactly when every finite critical point belongs to the filled Julia set. In this case the Böttcher coordinate extends conformally over the entire basin of infinity; an escaping critical point is precisely an obstruction to that continuation.

For $f_c(z)=z^2+c$ outside the Mandelbrot set, evaluate the dynamical Böttcher coordinate at the critical value:This parameter Böttcher map is a conformal isomorphism from the complement of the Mandelbrot set to the exterior unit disc, proving that the Mandelbrot set is full and connected.

The holomorphic fixed-point index isAt a simple fixed point of multiplier $\lambda \ne 1$, it equals $1/(1-\lambda )$. The indices of all fixed points of a rational map, counted appropriately, sum to one.

If a quadratic rational map has three distinct fixed points with multipliers ${\lambda }_0,{\lambda }_1,{\lambda }_2$, then ${\lambda }_i{\lambda }_j\ne 1$ for $i\ne j$. Otherwise the two corresponding indices already sum to one, contradicting the nonzero third index.

A parabolic cycle is a special kind of periodic point orbit.

A parabolic cycle is a periodic orbit whose multiplier is a root of unity; an iterate then has multiplier one and attracting petals.

For quadratic-polynomial parameter space, these are open components of the interior of the Mandelbrot set with an attracting periodic orbit.

A hyperbolic component in a parameter space consists of maps with a specified attracting cycle. Its multiplier gives a holomorphic coordinate on the component in the quadratic family, and root-of-unity boundary multipliers give parabolic parameters.

For every prime $p$, the polynomial $f_c^p(0)$ has a nonzero root, giving a superattracting cycle of exact period $p$. Moving in its hyperbolic component to a boundary point with cycle multiplier $-1$ gives a parabolic cycle of exact period $p$. Thus the set of parabolic periods in the quadratic family is infinite.