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<p>In fibred polynomial dynamics over an irrational rotation of the unit circle, invariant curves are the natural analog of fixed points. This paper addresses the existence of more complex minimal invariant sets. We introduce a new class of objects, called multi-curves, which are multi-component curves that wrap around the base space several times. Their analysis is conducted by lifting the dynamics to a covering space, where the multi-curve becomes a simple invariant curve. We present a constructive method, leveraging the monodromy of fixed points near a parabolic bifurcation in the Mandelbrot set, to create these objects. As a primary application of this method, we provide a constructive proof for the existence of an attracting invariant 2-curve within an explicit family of fibred quadratic polynomials. This result demonstrates a new phenomenon in fibred dynamics, showcasing a richer variety of invariant objects than in the classical one-dimensional setting and opening a new avenue for their classification and study.</p>

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On invariant multi-curves in fibred polynomial dynamics

  • 期刊论文

  • 发表日期：

    2026年2月26日

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  • Igsyl Domínguez

摘要

In fibred polynomial dynamics over an irrational rotation of the unit circle, invariant curves are the natural analog of fixed points. This paper addresses the existence of more complex minimal invariant sets. We introduce a new class of objects, called multi-curves, which are multi-component curves that wrap around the base space several times. Their analysis is conducted by lifting the dynamics to a covering space, where the multi-curve becomes a simple invariant curve. We present a constructive method, leveraging the monodromy of fixed points near a parabolic bifurcation in the Mandelbrot set, to create these objects. As a primary application of this method, we provide a constructive proof for the existence of an attracting invariant 2-curve within an explicit family of fibred quadratic polynomials. This result demonstrates a new phenomenon in fibred dynamics, showcasing a richer variety of invariant objects than in the classical one-dimensional setting and opening a new avenue for their classification and study.