2024/12/22 by Carsten Lunde Petersen, Petersen, Carsten Lunde, Saeed Zakeri +1 · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2412.17125
We introduce a general framework to study the local dynamics of near-parabolic maps using the meromorphic 1-form introduced by X.~Buff. As a sample application of this setup, we prove the following tameness result on invariant curves of near-parabolic maps: Let g(z)=λz+O(z2) have a non-degenerate parabolic fixed point at 0 with multiplier λ a primitive qth root of unity, and let γ: ]-∞,0] → \mathbb D(0,r) be a g∘ q-invariant curve landing at 0 in the sense that g∘ q(γ(t))=γ(t+1) and limt → -∞ γ(t)=0. Take a sequence gn(z)=λn z+O(z2) with |λn|≠ 1 such that gn → g uniformly on \mathbb D(0,r) and suppose each gn admits a gn∘ q-invariant curve γn: ]-∞,0] → \mathbb C such that γn → γ uniformly on the fundamental segment [-1,0]. If λnq → 1 non-tangentially, then γn lands at a repelling periodic point near 0, and γn → γ uniformly on ]-∞,0]. In the special case of polynomial maps, this proves Hausdorff continuity of external rays of a given periodic angle when the associated multipliers approach a root of unity non-tangentially.