The quadratic map $F(x,y)=(y,y^2-a-x)$ has a compact coded invariant set conjugate to the two-sign shift, with chaos under ${\mathrm{A}\mathrm{C}}_{\omega }$

Shift invariance of the code: fix $s\in X_{\pm }$ and define $y_n:=x_{n+1}(s)$ for $n\in \mathbb{Z}$. Then, using $T_{S_{\pm }s}$ and [F1], $(T_{S_{\pm }s}y{)}_n=(S_{\pm }s{)}_n\sqrt{a+y_{n-1}+y_{n+1}}=s_{n+1}\sqrt{a+x_n(s)+x_{n+2}(s)}=(T_{s}x(s){)}_{n+1}=x_{n+1}(s)=y_n;$ thus $y$ is a fixed point of $T_{S_{\pm }s}$ in $B_R$, and by the uniqueness clause of [F1] (via The square-root recurrence $T_s(x{)}_n=s_n\sqrt{a+x_{n-1}+x_{n+1}}$ is a contraction of $B_R$ with a unique coded solution) $y=x(S_{\pm }s)$, i.e. $x_n(S_{\pm }s)=x_{n+1}(s)$ for all $n$.

$F{\mathrm{\mid }}_{\mathrm{\Lambda }}$ is continuous: $F$ has polynomial components, hence is continuous on ${\mathbb{R}}^2$, and a restriction of a continuous map is continuous (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form).

Conjugacy identity: for $s\in X_{\pm }$, [F1] and step 1.1 give $F(h(s))=F(x_{-1}(s),x_0(s))=(x_0(s),x_1(s))=(x_{-1}(S_{\pm }s),x_0(S_{\pm }s))=h(S_{\pm }s)$. Hence $F(h(X_{\pm }))\subseteq h(X_{\pm })$, and since $S_{\pm }$ is onto, for every $h(s)\in h(X_{\pm })$ the identity applied to $S_{\pm }^{-1}s$ gives $h(s)=F(h(S_{\pm }^{-1}s))\in F(h(X_{\pm }))$; therefore $F(\mathrm{\Lambda })=\mathrm{\Lambda }$.

$h$ is a homeomorphism: $h$ is a continuous bijection of $X_{\pm }$ onto $\mathrm{\Lambda }$ by [F1]. To prove that $h^{-1}$ is continuous it suffices to prove that $h$ maps closed subsets of $X_{\pm }$ to closed subsets of ${\mathbb{R}}^2$. Let $C\subseteq X_{\pm }$ be closed. Then $C$ is sequentially compact: a sequence in $C$ has a subsequence converging in the sequentially compact space $X_{\pm }$ ([F2] and the isometry $\phi$ of [F1]), and the limit lies in $C$ because $C$ is closed and therefore sequentially closed by the choice-free direction of [F4]. The continuous image $h(C)$ is sequentially compact: given a sequence $(h(x_k))$ with $x_k\in C$, pick a subsequence $x_{k_j}$ converging to some $x\in C$, then $h(x_{k_j})\to h(x)\in h(C)$ by continuity (Countably compact, sequentially compact and limit point compact metric spaces, Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form); no choice is used beyond the single subsequence selection that sequential compactness of $C$ provides. Moreover $h(C)$ is sequentially closed: if a sequence in $h(C)$ converges, then some subsequence converges to a point of $h(C)$ by the previous paragraph (with that sequence in place of the given one), and the whole sequence has the same limit by uniqueness of limits ([F4]); hence the limit lies in $h(C)$. By the ${\mathrm{A}\mathrm{C}}_{\omega }$ direction of [F4] the sequentially closed set $h(C)$ is closed in ${\mathbb{R}}^2$. Thus $h$ is a closed map, so $h^{-1}$ is continuous and $h$ is a homeomorphism.

Sensitivity: let $z=h(s)\in \mathrm{\Lambda }$ and $\epsilon >0$; choose $m\ge 0$ with $4\sqrt{2}R{L}^{-m}<\epsilon$ and let $k\ge m+1$. Define $t\in X_{\pm }$ by $t_k:=-s_k$ and $t_j:=s_j$ for $j\mathrm{\ne }k$. Then $s,t$ agree for all $\mathrm{\mid }j\mathrm{\mid }\le m$, so by the coding-continuity clause of [F1] (Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h(\sigma )=(x_{-1}(\sigma ),x_0(\sigma ))$) one has $\mathrm{\parallel }h(s)-h(t){\mathrm{\parallel }}_2\le 4\sqrt{2}R{L}^{-m}<\epsilon$. On the other hand $S_{\pm }^{k}s$ and $S_{\pm }^{k}t$ differ exactly at coordinate $0$, so their signs at $0$ are opposite; by [F1], $\mid x_0(S_{\pm }^{k}s)-x_0(S_{\pm }^{k}t)\mid \ge \mathrm{\mid }{x}_0(S_{\pm }^{k}s)\mathrm{\mid }+\mathrm{\mid }{x}_0(S_{\pm }^{k}t)\mathrm{\mid }\ge 2L$. Using step 2.1, $F^k(z)=F^k(h(s))=h(S_{\pm }^{k}s)$ and $F^k(h(t))=h(S_{\pm }^{k}t)$, so $\parallel F^k(z)-F^k(h(t)){\parallel }_2\ge \mid x_0(S_{\pm }^{k}s)-x_0(S_{\pm }^{k}t)\mid \ge 2L$. Thus $2L$ is a sensitivity constant.

$\mathrm{\Lambda }$ is sequentially compact: $X_2$ is sequentially compact without choice ([F2]); since $\phi :X_{\pm }\to X_2$ is a bijective isometry ([F1]), a sequence in $X_{\pm }$ is carried to a sequence in $X_2$ with a convergent subsequence whose preimage under $\phi$ converges in $X_{\pm }$ to the corresponding preimage limit; hence $X_{\pm }$ is sequentially compact. Applying the continuous-image argument of step 3.1 with $C=X_{\pm }$ shows that $\mathrm{\Lambda }=h(X_{\pm })$ is sequentially compact.

$\mathrm{\Lambda }$ is compact under ${\mathrm{A}\mathrm{C}}_{\omega }+\mathrm{D}\mathrm{C}$ [F5]: $X_2$ is compact by [F2]; since $\phi$ is a bijective isometry, a set $U\subseteq X_{\pm }$ is open exactly when $\phi (U)\subseteq X_2$ is open, so the images under $\phi$ of an open cover of $X_{\pm }$ form an open cover of $X_2$, whose finite subcover pulls back to a finite subcover of the original cover; hence $X_{\pm }$ is compact. A continuous image of a compact space is compact by The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset, so $\mathrm{\Lambda }=h(X_{\pm })$ is compact.

Topological mixing of $F{\mathrm{\mid }}_{\mathrm{\Lambda }}$: let $U,V\subseteq \mathrm{\Lambda }$ be nonempty and open in $\mathrm{\Lambda }$; then $h^{-1}(U)$ and $h^{-1}(V)$ are nonempty open subsets of $X_{\pm }$ (step 3.1). Since $S_{\pm }$ is topologically mixing ([F2], transferred through the isometry $\phi$ of [F1] and the conjugacy $S_{\pm }={\phi }^{-1}\circ {\sigma }_2\circ \phi$), there is $N$ with $S_{\pm }^n(h^{-1}(U))\cap h^{-1}(V)\mathrm{\ne }\mathrm{\varnothing }$ for every $n\ge N$; applying $h$ and step 2.1, $F^n(U)\cap V=h\left(S_{\pm }^n\right(h^{-1}(U))\cap h^{-1}(V))\mathrm{\ne }\mathrm{\varnothing }$ for every $n\ge N$.

Dense periodic points: a point $s$ is periodic for $S_{\pm }$ if and only if $h(s)$ is periodic for $F{\mathrm{\mid }}_{\mathrm{\Lambda }}$, by the conjugacy of step 2.1 and the injectivity of $h$ ([F1]). Since $h$ is a bijection carrying the topology of $X_{\pm }$ onto that of $\mathrm{\Lambda }$ (steps 2.1, 3.1 and Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets), it maps the dense set of periodic points of $S_{\pm }$ ([F2]) to a dense subset of $\mathrm{\Lambda }$ consisting of periodic points of $F{\mathrm{\mid }}_{\mathrm{\Lambda }}$.

Conclusion: by steps 1.2, 4.3, 4.4 and 3.2, the restriction $F{\mathrm{\mid }}_{\mathrm{\Lambda }}$ is continuous, topologically mixing, has dense periodic points and is sensitive with constant $2L$; by the definition of Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list it is chaotic in the stated sense, and clauses 1-4 are proved.