= Repelling petal
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 $f(z)=z+az^{k+1}+O(z^{k+2})$, its inverse begins $f^{-1}(z)=z-az^{k+1}+O(z^{k+2})$. Repelling directions for $f$ therefore have $az^k$ 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.<Dynamics in one complex variable: introductory lectures by John Milnor>[Section 7, petal definition and Theorem 7.2]{ref}
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