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On the abundance of k-fold semi-monotone minimal sets in bimodal circle maps
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 5 / May 2024
- Published online by Cambridge University Press:
- 12 July 2023, pp. 1269-1314
- Print publication:
- May 2024
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- Article
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Inspired by a twist map theorem of Mather. we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map g to the k-fold cover. For each irrational in the rotation set’s interior, the collection of the k-fold ordered semi-Denjoy minimal sets with that rotation number contains a
$(k-1)$-dimensional ball with the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool used is a generalization of a construction of Hedlund and Morse that generates symbolic analogs of these k-fold well-ordered invariant sets.
