Abstract
Let \(S =\varGamma \setminus \mathcal{H}\) be a compact hyperbolic surface (or more generally, of finite area). In this last chapter we shall study the following conjecture.
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Notes
- 1.
The appearance of the word “quantum” here is explained in the first subsection.
- 2.
I.e., the corresponding set of gradient vectors is linearly independent at almost every point of \(\mathcal{H}\).
- 3.
The coordinates w and \(\overline{w}\) are natural here since we then have \(H = \frac{1} {2}w\overline{w}\).
- 4.
This paragraph is not needed for what comes after it. We shall freely use the ergodicity of the geodesic flow as well as the L 2 Birkhoff theorem, see [50].
- 5.
Note that this is the first time we are invoking the ergodicity of the geodesic flow.
- 6.
This type of property rigidifies the situation.
- 7.
We use here the term ergodic without entering into the details. See later sections or [50] for a definition.
- 8.
The fixed points of matrices in \(\mathrm{SL}(2,\mathbf{Z}\left [1/p\right ])\) form a countable set.
- 9.
Thus, even a precise control on the geodesic flow over an interval [0, T] cannot lead to an estimation on ρ(γ 1(2T), γ 2(2T)) (other than that implied by the triangle inequality). The future is independent of the past; this is a very simple manifestation of deterministic chaos.
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Bergeron, N. (2016). Arithmetic Quantum Unique Ergodicity. In: The Spectrum of Hyperbolic Surfaces. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-27666-3_9
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