Complex dynamics

Complex dynamics studies iterations of a map $z_{n+1}=f(z_n)$ when $f$ is a holomorphic map. A central case is a rational map of the Riemann sphere of degree of a rational map of the Riemann sphere at least two. One asks which orbits approach an attracting periodic point, which escape, and where nearby starting points have unstable long-term behavior.

The polynomial function $f(z)=z^2$ is a complete first example:For $|z|<1$ these iterations of a map tend to zero; for $|z|>1$ they tend to infinity on the Riemann sphere. On the unit circle, writing $z=e^{2\pi it}$ turns the iteration of a map into angle doubling $t\mapsto 2t(\mathrm{mod}1)$. The two open regions make up the Fatou set, where the iterations of a map form a normal family; their shared unit circle is the Julia set.

The quadratic polynomial family $f_c(z)=z^2+c$ illustrates two different questions. For a fixed $c$, its Julia set lives in the dynamical $z$-plane. The Mandelbrot set lives in the parameter $c$-plane and records when the critical orbit starting at zero stays bounded. Varying a parameter and varying an initial point are separate operations.

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 $\mathcal{F}$ of meromorphic functions on a domain is a normal family if every sequence from $\mathcal{F}$ has a subsequence converging uniformly on every compact subset, measured in the chordal metric on the Riemann sphere. The limit is a meromorphic function or the constant infinity. For holomorphic functions, a finite limit is holomorphic; a sequence may also tend locally uniformly to infinity.

The constant functions $f_n(z)=n$ form a normal family, because any sequence has either a constant subsequence or a subsequence tending uniformly to infinity in the chordal metric. In contrast $g_n(z)=nz$ is not a normal family on any neighbourhood of zero. Every subsequence tends to infinity at every nonzero point but remains zero at zero; this pointwise pattern cannot be a continuous Riemann sphere valued limit.

For a rational map $f$, the Fatou set is precisely where its iterations of a map form a normal family. Normality asks for convergent subsequences, so it allows stable rotations as well as convergence to an attracting periodic point.^[ref]

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.

Suppose a family of meromorphic functions is a normal family on a neighbourhood of every point of a planar domain. Choose a countable cover by open discs with that property. From any sequence extract successive subsequences converging locally uniformly in the chordal metric on the first disc, second disc, and so on; take the diagonal subsequence argument.

The local Riemann sphere valued limits agree on overlaps, since the same subsequence converges there. They therefore define one continuous global limit. Locally it is meromorphic or identically infinity, so on a connected domain it is globally meromorphic or identically infinity. A finite cover of each compact subset transfers the local uniform estimates to that subset. Thus the original family is a normal family on the entire domain. The argument handles poles using the chordal metric; it does not require every local limit to be a finite holomorphic function.

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 $F(f)$ of a rational map is the set of points having a neighbourhood on which the iterations of a map $\{f^n{\}}_{n\ge 0}$ form a normal family. It is open by definition, and its connected components are the Fatou components. Its complement on the Riemann sphere is the Julia set.

For $f(z)=z^2$, the Fatou set is $\{|z|<1\}\cup \{|z|>1\}\cup \{\infty \}$. The iterations of a map converge locally uniformly to zero inside the unit disc and to infinity outside its closure. Each region is an attracting basin. More generally the Fatou set also contains rotation domains such as a Siegel disc: stability in the normal-family sense does not require convergence to one point.

The Fatou set is a completely invariant set of a rational map, so $f^{-1}(F(f))=F(f)=f(F(f))$. A single Fatou component may instead move to another component under $f$.^[ref]

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.

With the closed-set convention, the postcritical set of a rational map $f$ iswhere $\mathrm{Crit}(f)$ is the set of critical points of a rational map. Thus one starts at the critical values, takes all their forward orbits, and then takes the closure. Some authors omit the closure; stating the convention matters when a critical orbit accumulates without becoming periodic.

For $f(z)=z^2-1$, the critical points of a rational map are zero and infinity. Their forward images give $P(f)=\{-1,0,\infty \}$, because $0\mapsto -1\mapsto 0$ and infinity is fixed. The postcritical set combines all critical orbits, rather than recording only one chosen starting point.

A parabolic basin consists of points that eventually enter an attracting direction of a parabolic cycle. For a suitable return iterate, an attracting petal is a forward-invariant open region with the periodic point on its boundary, and the return iterates tend to that point there. The union of its inverse images is an open parabolic basin contained in the Fatou set; the Fatou components meeting the attracting petals form the immediate basin. Its boundary lies in the Julia set.

The parabolic cycle itself and points that land exactly on it belong to the Julia set, so mere pointwise convergence to that cycle is not the definition of the open parabolic basin. For $f(z)=z+z^2$, zero is a fixed point with periodic-orbit multiplier one. Small negative real starting values approach zero without reaching it, whereas zero itself is a Julia set point. The reflected example parabolic disk for z minus z squared gives an explicit forward-invariant region and a quantitative convergence bound.^[ref]

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.

For a rational map $f$ of degree of a rational map of the Riemann sphere at least two, the Julia set isthe points where the iterations of a map fail to form a normal family on every neighbourhood. It is nonempty, closed, a completely invariant set of a rational map, and perfect. The repelling periodic points are dense in the Julia set.

For $f(z)=z^2$, the Julia set is the unit circle. Every neighbourhood of a point on that circle meets points whose iterations of a map tend to zero and points whose iterations of a map tend to infinity. These two incompatible local limits prevent normality there. The circle separates the two Fatou components.

A Julia set need not enclose a nonempty interior region: the Chebyshev polynomial Julia set is an interval, and other rational maps have a Julia set equal to the whole Riemann sphere. For a polynomial function, distinguish the Julia set from the filled Julia set, which includes every nonescaping point and has the Julia set as its boundary.^[ref]

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 $|E|\ge 3$.

Let $f$ be a rational map of degree of a rational map of the Riemann sphere at least two. Its nonempty Julia set $J$ is infinite: a finite completely invariant closed set of a rational map lies in the Fatou set. Since $J$ is compact, its set $J^{\prime }$ of accumulation points is nonempty. It is closed, contained in $J$, and completely invariant; local openness of a nonconstant holomorphic map gives the inverse-image direction.

Apply the completely invariant closed set of a rational map classification to $J^{\prime }$. Its alternative with at most two points would put the nonempty set $J^{\prime }$ in the Fatou set, contradicting $J^{\prime }\subseteq J$. Hence $J\subseteq J^{\prime }$, so $J=J^{\prime }$ and the Julia set has no isolated points. This supplies the perfect property without assuming it in advance.^[ref]

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 $1/(1-\lambda )$. At infinity, use a local coordinate centred there and the corresponding local expression for the map. For a rational map other than the identity, all fixed points are isolated and their holomorphic fixed-point indices 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.

For a rational map of degree of a rational map of the Riemann sphere at least two, a parabolic cycle is a cycle of a holomorphic map whose periodic-orbit multiplier is a root of unity. If the exact period is $p$ and the multiplier has order $q$, the return iterate $f^{pq}$ has multiplier one. In a local coordinate $u$ centred at a point of the cycle it has the formThe nonzero nonlinear term gives attracting petals and repelling petals, with the former underlying a parabolic basin. A multiplier of modulus one with irrational argument is not parabolic.

For $f_c(z)=z^2+c$ at $c=1/4$, the fixed point $z=1/2$ has multiplier one. The translation $z=1/2+u$ gives the exact local map $u\mapsto u+u^2$, exposing the nonlinear term responsible for attraction from one direction and repulsion from another. The parabolic cycle belongs to the Julia set even though nearby points in its parabolic basin approach it.^[ref]

Suppose a holomorphic function near zero has expansionThe Leau-Fatou flower theorem gives $k$ attracting petals for forward iteration and $k$ repelling petals for the local inverse, alternating around the parabolic fixed point. They can be chosen so that together with zero they cover a neighborhood. Within an attracting petal, every forward orbit tends to zero along its attracting direction.^[ref]

The change of variable $w=-1/(ka{z}^k)$ explains the geometry: substitution gives $w(f(z))=w(z)+1+o(1)$. Iteration therefore behaves like translation far out in a half-plane, and the $k$ inverse branches produce the $k$ petals. In particular $z_n^k\sim -1/(kan)$ along an attracting orbit. For $f(z)=z+z^2$, small negative real points approach zero like $-1/n$, while small positive real points move away. A derivative equal to one hides dynamics that the first nonlinear term reveals.

A fixed point $z_0$ of a holomorphic map is parabolic when its multiplier $f^{\prime }(z_0)$ is a root of unity and the corresponding return iterate is not locally the identity. In the usual rational map setting of degree of a rational map of the Riemann sphere at least two, the latter exclusion is automatic. If the multiplier has order $q$, then $f^q$ has derivative one at $z_0$ and a nonzero higher-order term.

For $f(z)=z^2+1/4$, the point $z_0=1/2$ is fixed and $f^{\prime }(z_0)=1$. Writing $z=1/2+u$ gives exactly $u\mapsto u+u^2$. Small negative real $u$ approach zero, whereas small positive real $u$ move away. The Leau-Fatou flower theorem describes the corresponding sectors in the complex plane. This behavior differs from uniform local attraction with multiplier of modulus less than one, and from an irrational rotation multiplier of modulus one.^[ref]

An attracting petal is a forward-invariant region approaching a parabolic fixed point of a holomorphic map. For a local map fixing zero with multiplier one,one can choose a small connected open region $U$, with zero on its boundary, such that $f(\overline{U})\subseteq U\cup \{0\}$ and the iterated images of $\overline{U}$ shrink to zero. In particular, every orbit starting in $U$ tends to the fixed point.

The Leau-Fatou flower theorem gives $k$ attracting petals, alternating with $k$ repelling petals. The leading directions are determined by $a{z}^k$ being negative real. For $f(z)=z+z^2$, negative real points sufficiently close to zero move monotonically toward zero; the attracting petal surrounds that direction. This attraction comes from the nonlinear term, since the derivative at the fixed point has modulus one.

For a parabolic cycle whose multiplier is another root of unity, the petal construction is applied to a suitable return iterate with multiplier one. Points whose orbits eventually enter an attracting petal form its parabolic basin.^[ref]

A repelling petal for a holomorphic map at a parabolic fixed point is an attracting petal for its local inverse. The inverse exists near the fixed point because the multiplier is nonzero. Backward iterates remain in the petal and approach the fixed point, whereas a nonzero point followed forward eventually leaves the small repelling region.

For $f(z)=z+a{z}^{k+1}+O(z^{k+2})$, its inverse begins $f^{-1}(z)=z-a{z}^{k+1}+O(z^{k+2})$. Repelling directions for $f$ therefore have $a{z}^k$ positive real, alternating with the attracting directions. The Leau-Fatou flower theorem provides $k$ petals of each kind.

For $f(z)=z+z^2$, small positive real points increase under forward iteration, while the inverse branch $f^{-1}(z)=(-1+\sqrt{1+4z})/2$ moves them toward zero. This verifies the direction of repulsion without treating multiplier one as an expanding linear map.^[ref]

In the quadratic polynomial family $f_c(z)=z^2+c$, a hyperbolic component is a connected component of the set of parameters for which $f_c$ has an attracting periodic point. The finite critical orbit is then attracted to that cycle. Its exact period is constant throughout the component. The component is open because an attracting periodic point and its strict multiplier inequality persist under small parameter changes.

The main cardioid has an attracting periodic point of period one; the period-two bulb has an attracting exact two-cycle. Their periodic-orbit multipliers give the explicit coordinates $c=\lambda /2-{\lambda }^2/4$ and $\lambda =4(c+1)$, respectively, with $|\lambda |<1$. More generally the attracting-cycle multiplier gives a conformal coordinate on each quadratic polynomial hyperbolic component.

This is a definition within the specified parameter family. For general rational maps, the existence of one attracting cycle alone does not require every critical orbit to approach an attracting cycle, so it does not by itself establish global hyperbolicity.^[ref]

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.