Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family

The quadratic family. For $a\in \mathbb{R}$ let

$F_a:{\mathbb{R}}^2⟶{\mathbb{R}}^2,F_a(x,y):=(y,y^2-a-x),$

with differential at $(x,y)\in {\mathbb{R}}^2$

$D{F}_a(x,y)=\left(\begin{array}{cc}0& 1\\ -1& 2y\end{array}\right)\mathrm{.}$

The inverse map is

$F_a^{-1}(x,y)=(x^2-a-y,x),$

and its differential at $(x,y)$ is

$D(F_a^{-1})(x,y)=\left(\begin{array}{cc}2x& -1\\ 1& 0\end{array}\right)\mathrm{.}$

Here $D{F}_a(x,y{)}^{-1}$ denotes the inverse matrix of $D{F}_a(x,y)$, namely $\left(\begin{array}{cc}2y& -1\\ 1& 0\end{array}\right)$; it is not the same notation as $D(F_a^{-1})(x,y)$, which evaluates the derivative of the inverse map at the point $(x,y)$. All matrices act on column vectors of ${\mathbb{R}}^2$ (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

Complex coordinates and modulus. Use $\mathbb{C}:=\mathbb{R}[x]\mathrm{/}(x^2+1)$, with real numbers embedded as constant classes and $i:=x+(x^2+1)$, so $i^2=-1$ (The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$). This is a field and each element has a unique form $\alpha +i\beta$ with $\alpha ,\beta \in \mathbb{R}$ ($\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$). Define its modulus by

$\mathrm{\mid }\alpha +i\beta \mathrm{\mid }:=\sqrt{{\alpha }^2+{\beta }^2},$

where the square root is the unique nonnegative real square root (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$). Identify ${\mathbb{C}}^2$ with pairs of complex coordinates; a real matrix acts on it by the same matrix entries, extended $\mathbb{C}$-linearly.

Hyperbolic fixed points. Let $U,V\subseteq {\mathbb{R}}^2$ be open, let $G:U\to V$ be a $C^1$ diffeomorphism, and let $\bar{z}\in U\cap V$ satisfy $G(\bar{z})=\bar{z}$. Its eigenvalues are the numbers $\lambda \in \mathbb{C}$ for which $DG(\bar{z})v=\lambda v$ for some $v\in {\mathbb{C}}^2\setminus \{0\}$. The fixed point $\bar{z}$ is hyperbolic when every such eigenvalue satisfies $\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\ne }1$. A global $C^1$ diffeomorphism of ${\mathbb{R}}^2$ is the special case $U=V={\mathbb{R}}^2$.

Cones. Fix $k\in (0,1)$. The unstable cone and stable cone of aperture $k$ are

$C_k^u:=\{(u,v)\in {\mathbb{R}}^2:\mathrm{\mid }u\mathrm{\mid }\le k\mathrm{\mid }v\mathrm{\mid }\},C_k^s:=\{(u,v)\in {\mathbb{R}}^2:\mathrm{\mid }v\mathrm{\mid }\le k\mathrm{\mid }u\mathrm{\mid }\}$

(Absolute value in an ordered field); both are closed subsets of ${\mathbb{R}}^2$, they are closed under multiplication by real scalars, and $C_k^u\cap C_k^s=\{0\}$, because a nonzero vector in the intersection would satisfy $\mathrm{\mid }u\mathrm{\mid }\le k\mathrm{\mid }v\mathrm{\mid }$ and $\mathrm{\mid }v\mathrm{\mid }\le k\mathrm{\mid }u\mathrm{\mid }$, hence $\mathrm{\mid }u\mathrm{\mid }\le k^2\mathrm{\mid }u\mathrm{\mid }<\mathrm{\mid }u\mathrm{\mid }$.

Slopes. A line through the origin not contained in the horizontal axis is written uniquely as $\{(hv,v):v\in \mathbb{R}\}$ with the slope parameter $h=u\mathrm{/}v$; here $\mathrm{\mid }h\mathrm{\mid }\le k$ characterises the lines contained in $C_k^u$. Similarly a line not contained in the vertical axis is written uniquely as $\{(u,ru):u\in \mathbb{R}\}$ with the slope parameter $r=v\mathrm{/}u$, and $\mathrm{\mid }r\mathrm{\mid }\le k$ characterises the lines contained in $C_k^s$.

For real numbers $x,y$ two maps on slope parameters are defined:

${\mathrm{\Phi }}_y(h):=\frac{1}{2y-h}(h\mathrm{\ne }2y)\text{(forward slope map)},{\mathrm{\Psi }}_x(r):=\frac{1}{2x-r}(r\mathrm{\ne }2x)\text{(backward slope map)}\mathrm{.}$

The intended reading is the graph-transform reading: $D{F}_a(x,y)$ carries the line $\{(hv,v):v\in \mathbb{R}\}$ at $(x,y)$ to the line $\{(h^{\mathrm{\prime }}v,v):v\in \mathbb{R}\}$ at $F_a(x,y)$ with $h^{\mathrm{\prime }}={\mathrm{\Phi }}_y(h)$, and $D(F_a^{-1})(x,y)$ carries the line $\{(u,ru):u\in \mathbb{R}\}$ at $(x,y)$ to the line $\{(u,r^{\mathrm{\prime }}u):u\in \mathbb{R}\}$ at $F_a^{-1}(x,y)$ with $r^{\mathrm{\prime }}={\mathrm{\Psi }}_x(r)$. These two statements follow from the displayed matrices and are proved in the companion lemma on this page, not asserted as part of this definition.

Invariant line fields on a sequence. A real bilateral sequence is a function $s:\mathbb{Z}\to \mathbb{R}$; write $s_n:=s(n)$. Put $z_n:=(s_{n-1},s_n)$. An unstable line field along $s$ is a family $(L_n^u{)}_{n\in \mathbb{Z}}$ of lines through the origin of ${\mathbb{R}}^2$ such that $D{F}_a(z_n)$ maps $L_n^u$ onto $L_{n+1}^u$ for every $n$. A stable line field along $s$ is a family $(L_n^s{)}_{n\in \mathbb{Z}}$ such that $D(F_a^{-1})(z_{n+1})$ maps $L_{n+1}^s$ onto $L_n^s$ for every $n$. The latter evaluates the derivative of the inverse map at $z_{n+1}$. When $s$ satisfies $s_{n+1}=s_n^2-a-s_{n-1}$, one has $z_{n+1}=F_a(z_n)$, so these are the source and target orbit points. The displayed matrix conditions also define line families for arbitrary sequences, without asserting that such a sequence is an orbit. For a real bound $L\ge 2$, sequences with $\mathrm{\mid }{s}_n\mathrm{\mid }\ge L$, and apertures satisfying $k(2L-k)\ge 1$, existence and uniqueness of line fields with slopes in $[-k,k]$ are proved in the companion lemma on this page.

Stable and unstable sets. Let $G:Z\to Z$ be a bijection of a metric space $(Z,d)$ (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric) with fixed point $\bar{z}$ (so $G(\bar{z})=\bar{z}$). Put

$W^s(\bar{z}):=\{z\in Z:G^n(z)\text{ is defined for every }n\ge 0\text{ and }{G}^n(z)\to \bar{z}\text{ as }n\to \mathrm{\infty }\},$

$W^u(\bar{z}):=\{z\in Z:G^{-n}(z)\text{ is defined for every }n\ge 0\text{ and }{G}^{-n}(z)\to \bar{z}\text{ as }n\to \mathrm{\infty }\},$

where convergence is that of the metric $d$ (Convergence of a sequence in a metric space: $x_k\to x$ iff $d(x_k,x)\to 0$ in $\mathbb{R}$). The sets $W^s(\bar{z})$ and $W^u(\bar{z})$ are the stable set and the unstable set of $\bar{z}$.

These orbit-based definitions also apply to a local map. Let $U,V\subseteq {\mathbb{R}}^2$ be open, let $G:U\to V$ be a $C^1$ diffeomorphism, and let $\bar{z}\in U\cap V$ satisfy $G(\bar{z})=\bar{z}$. Define $G^0(z):=z$ and recursively $G^{n+1}(z):=G(G^n(z))$ whenever $G^n(z)\in U$; define $G^{-n-1}(z):=G^{-1}(G^{-n}(z))$ whenever $G^{-n}(z)\in V$. Then

$W_G^s(\bar{z}):=\{z\in U:G^n(z)\text{ is defined for every }n\ge 0\text{ and }{G}^n(z)\to \bar{z}\},$

$W_G^u(\bar{z}):=\{z\in V:G^{-n}(z)\text{ is defined for every }n\ge 0\text{ and }{G}^{-n}(z)\to \bar{z}\},$

where convergence is Euclidean metric convergence. The explicit domain clauses require every forward iterate to lie in $U$ before applying $G$, and every backward iterate to lie in $V$ before applying $G^{-1}$. In the global case above every iterate is defined automatically.

A homoclinic point is a point in both the stable and unstable sets, different from the fixed point $\bar{z}$. For a global bijection on ${\mathbb{R}}^2$ or a local planar diffeomorphism as above, it is transverse when there are $\epsilon >0$ and $C^1$ maps ($C^k$ Euclidean maps and diffeomorphisms, The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set)

${\gamma }_s,{\gamma }_u:(-\epsilon ,\epsilon )⟶{\mathbb{R}}^2$

such that ${\gamma }_s(0)={\gamma }_u(0)=z$, the stable curve takes values in the stable set and the unstable curve takes values in the unstable set, both are injective on $(-\epsilon ,\epsilon )$ with ${\gamma }_s^{\mathrm{\prime }}(0)\mathrm{\ne }0\mathrm{\ne }{\gamma }_u^{\mathrm{\prime }}(0)$, and the derivative vectors ${\gamma }_s^{\mathrm{\prime }}(0),{\gamma }_u^{\mathrm{\prime }}(0)\in {\mathbb{R}}^2$ are linearly independent. Thus derivatives and tangent vectors in this definition are taken in the two-dimensional differentiable setting used by the results on this page; no derivative is assigned to a curve in an arbitrary metric space. The interval $(-\epsilon ,\epsilon )$ is that of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length.

• The definition records the two vocabularies used by the coding theorem and the transversality theorem: the linear cone and slope objects of the quadratic family, and the invariant-set objects of global bijections and local planar diffeomorphisms. It asserts no invariance of the cones, no convergence of the slope maps and no transversality; those are the contents of Cone invariance and expansion, and convergence of the slope graph transforms to unique invariant stable and unstable line fields and The coded curves are genuine $C^1$ stable and unstable curves of $F$, with transverse intersection at every finite-word homoclinic point.

• The transversality condition above is stated with two $C^1$ curves through the point, one lying in each invariant set, whose tangents span ${\mathbb{R}}^2$. When the local stable and unstable sets are themselves $C^1$ curves near $z$ (which holds at the fixed points and their finite images in the situation of The coded curves are genuine $C^1$ stable and unstable curves of $F$, with transverse intersection at every finite-word homoclinic point), this agrees with the usual formulation "$W^s$ and $W^u$ meet transversely at $z$".

• The forward slope map is defined on $[-k,k]$ when $\mathrm{\mid }y\mathrm{\mid }>k$, since $\mathrm{\mid }2y-h\mathrm{\mid }\ge 2\mathrm{\mid }y\mathrm{\mid }-k>0$; this condition alone does not ensure contraction. In Cone invariance and expansion, and convergence of the slope graph transforms to unique invariant stable and unstable line fields, the stronger bound $\mathrm{\mid }y\mathrm{\mid }\ge L\ge 2$ gives $\mathrm{\mid }{\mathrm{\Phi }}_y(h)-{\mathrm{\Phi }}_y(h^{\mathrm{\prime }})\mathrm{\mid }\le (2L-k{)}^{-2}\mathrm{\mid }h-h^{\mathrm{\prime }}\mathrm{\mid }$ for $h,h^{\mathrm{\prime }}\in [-k,k]$, with $(2L-k{)}^{-2}<1$. The additional aperture condition $k(2L-k)\ge 1$ ensures that the image lies in $[-k,k]$. The same conclusions hold for the backward slope map with $\mathrm{\mid }x\mathrm{\mid }\ge L$.