Buff forms and invariant curves of near-parabolic maps
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| Format: | Preprint |
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2024
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| _version_ | 1866916538721763328 |
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| author | Petersen, Carsten Lunde Zakeri, Saeed |
| author_facet | Petersen, Carsten Lunde Zakeri, Saeed |
| contents | 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 $g(z)=λz+O(z^2)$ have a non-degenerate parabolic fixed point at $0$ with multiplier $λ$ a primitive $q$th root of unity, and let $γ: \, ]-\infty,0] \to {\mathbb D}(0,r)$ be a $g^{\circ q}$-invariant curve landing at $0$ in the sense that $g^{\circ q}(γ(t))=γ(t+1)$ and $\lim_{t \to -\infty} γ(t)=0$. Take a sequence $g_n(z)=λ_n z+O(z^2)$ with $|λ_n|\neq 1$ such that $g_n \to g$ uniformly on ${\mathbb D}(0,r)$ and suppose each $g_n$ admits a $g_n^{\circ q}$-invariant curve $γ_n: \, ]-\infty,0] \to {\mathbb C}$ such that $γ_n \to γ$ uniformly on the fundamental segment $[-1,0]$. If $λ_n^q \to 1$ non-tangentially, then $γ_n$ lands at a repelling periodic point near $0$, and $γ_n \to γ$ uniformly on $]-\infty,0]$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_17125 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Buff forms and invariant curves of near-parabolic maps Petersen, Carsten Lunde Zakeri, Saeed Dynamical Systems 37F10, 37F20, 37F40 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 $g(z)=λz+O(z^2)$ have a non-degenerate parabolic fixed point at $0$ with multiplier $λ$ a primitive $q$th root of unity, and let $γ: \, ]-\infty,0] \to {\mathbb D}(0,r)$ be a $g^{\circ q}$-invariant curve landing at $0$ in the sense that $g^{\circ q}(γ(t))=γ(t+1)$ and $\lim_{t \to -\infty} γ(t)=0$. Take a sequence $g_n(z)=λ_n z+O(z^2)$ with $|λ_n|\neq 1$ such that $g_n \to g$ uniformly on ${\mathbb D}(0,r)$ and suppose each $g_n$ admits a $g_n^{\circ q}$-invariant curve $γ_n: \, ]-\infty,0] \to {\mathbb C}$ such that $γ_n \to γ$ uniformly on the fundamental segment $[-1,0]$. If $λ_n^q \to 1$ non-tangentially, then $γ_n$ lands at a repelling periodic point near $0$, and $γ_n \to γ$ uniformly on $]-\infty,0]$. 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. |
| title | Buff forms and invariant curves of near-parabolic maps |
| topic | Dynamical Systems 37F10, 37F20, 37F40 |
| url | https://arxiv.org/abs/2412.17125 |