Horn maps of holomorphic functions locally pseudo-conjugate on their parabolic basins

The lifted horn map of a holomorphic function with a simple parabolic point is well known to be a complete local conjugacy invariant; this is a classical result proved independently by \'Ecalle, Voronin, Martinet and Ramis. Lanford and Yampolski have shown that, if two functions $f_1, f_2$ with simple parabolic points at $z_1, z_2$ are globally conjugate on their immediate parabolic basins, with the conjugacy and its inverse continuous at $z_1$, resp. $z_2$, then their horn maps must be cover-equivalent: there are isomorphisms $\psi^+ : \mathcal{D}_1^+\to \mathcal{D}_2^+$ and $\psi^- : \mathcal{D}_1^-\to \mathcal{D}_2^-$ between the top and bottom connected components of their domains, and a translation $T$ on the cylinder, such that $\mathbb{h}_2\circ\psi^+ = T\circ \mathbb{h}_1$ and $\mathbb{h}_2\circ\psi^- = T\circ \mathbb{h}_1$ holds on these domains. In this article, we introduce a notion of (semi) local conjugacy on immediate parabolic basins, which we call local pseudo-conjugacy and which in particular does not make any continuity assumption, and show that the horn maps $\mathbb{h}_1$ and $\mathbb{h}_2$ satisfy the condition above if and only if the two functions $f_1, f_2$ are locally pseudo-conjugate. This result is a first step to better understand invariant classes by parabolic renormalization.