The constant-code slope transform at $a=100$: the invariant slope of $h\mapsto 1\mathrm{/}(2r_{+}-h)$ and the cone constants

Given: The parameter $a=100$, the constants $L=\sqrt{74}$, $k=1\mathrm{/}2$, $r_{+}=1+\sqrt{101}$, and the slope maps of Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family.

Slope transform data (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family, Cone invariance and expansion, and convergence of the slope graph transforms to unique invariant stable and unstable line fields): for $L\ge 2$, $0<k<1$, $k(2L-k)\ge 1$, $\mathrm{\mid }y\mathrm{\mid }\ge L$ and $\mathrm{\mid }h\mathrm{\mid },\mathrm{\mid }{h}^{\mathrm{\prime }}\mathrm{\mid }\le k$, the forward slope map satisfies $\mathrm{\mid }{\mathrm{\Phi }}_y(h)\mathrm{\mid }\le (2L-k{)}^{-1}$ and $\mathrm{\mid }{\mathrm{\Phi }}_y(h)-{\mathrm{\Phi }}_y(h^{\mathrm{\prime }})\mathrm{\mid }\le (2L-k{)}^{-2}\mathrm{\mid }h-h^{\mathrm{\prime }}\mathrm{\mid }$; under these hypotheses the cone $C_k^u$ is invariant under $DF$ at points with second coordinate of absolute value at least $L$, with expansion factor $2L-k>1$.

Constant-code check using 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, clause 4: for the given $r_{+}=1+\sqrt{101}$, direct expansion gives $r_{+}^2=100+2r_{+}$ and $0<r_{+}<12<13=R$. Thus the constant sequence $x_n=r_{+}$ lies in $B_R$ and satisfies $r_{+}=\sqrt{100+2r_{+}}$; uniqueness of the coded fixed point identifies it with the all-$+1$ code.

Square roots and inequalities (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$, Absolute value in an ordered field, Basic properties of the absolute value): $r_{+}>10$ because $\sqrt{101}>10$; $\sqrt{r_{+}^2-1}>r_{+}-1$; $2\sqrt{74}>17$ because $4\cdot 74=296>289={17}^2$.