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arXiv · 2604.09441

Infinitely Many Attracting Periodic Circles in Higher Dimensions

Shuntaro Tomizawa
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Abstract

We study $C^r$ ($5 \le r \le \infty$) diffeomorphisms on closed manifolds of dimension at least three with a heteroclinic cycle between two hyperbolic periodic points. At each point, the unstable direction is one dimensional, and the stable and unstable eigenvalues closest to $1$ in modulus are real and simple. One heteroclinic connection is transverse and the other is non-transverse, and the product of those two eigenvalues is less than $1$ at one point and greater than $1$ at the other. Arbitrarily close to such a map, there are open sets in which a residual subset of diffeomorphisms has infinitely many attracting normally hyperbolic periodic circles. The proof uses a rescaling to the standard Hénon map and a corrected formula for the Lyapunov coefficient on its Neimark-Sacker (Andronov-Hopf) line.

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Shuntaro Tomizawa. 2026-04-10. Infinitely Many Attracting Periodic Circles in Higher Dimensions. https://arxiv.org/abs/2604.09441

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