Abstract
Throughout this chapter d will be a fixed integer ≥ 2. We study certain invariant sets for the multiplication by d map \(m_d: {\mathbb T} \to {\mathbb T}\) defined by
The low-degree cases m 2 and m 3 are often referred to as the doubling and tripling maps.
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- 1.
Thus, our notion of invariance is stronger than forward invariance m d (X) ⊂ X and weaker than full invariance \(m_d^{-1}(X)=X\).
- 2.
Assuming \(m_d^{-1}(E)=E\) for some measurable set E, the characteristic function χ E satisfies χ E ∘ m d = χ E . Expanding χ E into the Fourier series \(\sum c_n e^{2 \pi i n t}\), it follows that \(\sum c_n e^{2 \pi i d n t} = \sum c_n e^{2 \pi i n t}\) which implies c n = c dn for all n. Since c n → 0, this can hold only if c n = 0 for all n ≠ 0.
- 3.
He proves the statement for the closure of the union of all finite rotation sets for m d , but an inspection of his proof shows that it also works for the a priori larger set \({\mathcal R}_d\). The zero dimension statement for individual rotation sets was known much earlier [29].
- 4.
In fact, it will follow from the results of this section that for rotation sets minimality is equivalent to having a single dense orbit, a property that is often called point transitivity.
- 5.
The terminology is meant to suggest that nothing can be added to or removed from such a set without losing the property of being a rotation set.
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Zakeri, S. (2018). Rotation Sets. In: Rotation Sets and Complex Dynamics. Lecture Notes in Mathematics, vol 2214. Springer, Cham. https://doi.org/10.1007/978-3-319-78810-4_2
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DOI: https://doi.org/10.1007/978-3-319-78810-4_2
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