Isolated fixed points need not be nondegenerate

Isolatedness of a fixed point is strictly weaker than nondegeneracy. A fixed point is nondegenerate when $I-D{f}_x$ is invertible (Nondegenerate fixed point), whereas the local fixed point index of Isolated fixed point and local fixed point index is defined for every isolated fixed point, including degenerate ones. The equivalence of nondegeneracy with transversality of the graph to the diagonal is Graph-diagonal transversality is exactly fixed-point nondegeneracy, and it is exactly this transversality that fails at a degenerate isolated point.

The standard example. Take the local model $f(z)=z+z^2$ on $\mathbb{C}$, a smooth self-map of the plane ($C^r$ and smooth maps between smooth manifolds). Its fixed point equation is $z+z^2=z$, i.e. $z^2=0$, so $0$ is the only fixed point near the origin and it is isolated. Its differential is $D{f}_0=I$ and $I-D{f}_0=0$ is not invertible (Invertible linear maps, linear isomorphisms, and inverse linear maps), so $0$ is isolated but degenerate, and the determinant formula of The index of a nondegenerate fixed point is the sign of det(I-Df) does not apply.

Its index is nevertheless defined and equals $2$. With the convention $I-Df$ of this page the displacement is $\mathrm{i}\mathrm{d}-f=-z^2$, whose representative in real coordinates on the circle $\mathrm{\mid }z\mathrm{\mid }=\epsilon$ is $v\mapsto -{\epsilon }^2{e}^{2i\theta }$ with $\theta$ the polar angle; after normalization this is the self-map $e^{i\theta }\mapsto e^{i(2\theta +\pi )}$ of $S^1$, of degree $2$. Hence ${\mathrm{ind}}_0(f)=2$: an isolated degenerate fixed point can carry a nonzero index, and its value is not controlled by $I-D{f}_0$.

The two theorems on this page that survive. The definition of the local index applies verbatim, and under countable choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), An isolated fixed point splits under perturbation, preserving its index splits the degenerate point into nondegenerate ones with the same total index. For the explicit quadratic perturbation $f_a(z)=z+z^2-a$, $a\in \mathbb{C}\setminus \{0\}$ small, the two fixed points satisfy $z^2=a$. At either point the displacement derivative is multiplication by $-2z$, of real determinant $4\mathrm{\mid }z{\mathrm{\mid }}^2>0$, so each index is $+1$. General smooth perturbations can have more fixed points; the splitting theorem preserves the total index, not their number. The polynomial extends to a smooth self-map of the Riemann sphere $S^2=\mathbb{C}\cup \{\mathrm{\infty }\}$ whose only fixed points are $0$ and $\mathrm{\infty }$: in the chart $w=1\mathrm{/}z$ the map is $w\mapsto w^2\mathrm{/}(w+1)$ and the fixed point equation $w^2\mathrm{/}(w+1)=w$ has the unique solution $w=0$, with displacement $w-w^2\mathrm{/}(w+1)=w\mathrm{/}(w+1)$, whose linear part at $0$ is the identity, so ${\mathrm{ind}}_{\mathrm{\infty }}(f)=+1$. This is the standard example showing that the converse direction of the Lefschetz theory needs the index and not merely the first derivative; the companion examples page computes both indices.