Quantum ergodicity (QE) is the study of behavior of eigenstates of a quantized Hamiltonian system in the so-called semiclassical limit, corresponding to large energies, or equivalently to the quantization parameter \(\hbar\rightarrow0\). If \(f\) is a classical observable, \(\text{Op}_{\hbar_{k}}(f)\) denotes the quantization of \(f\) with respect to \(\hbar=\hbar_{k}\) (\(k\geq1\)) and \(\psi_{k}\) are the eigenstates of the corresponding quantized Hamiltonian, then there is an associated measure \(\mu_{k}(f)=(\psi_{k},\text{Op}_{\hbar_{k}}(f)\psi)\). In QE, we are then interested in the possible measures \(\mu\) which can appear as limits \(\mu=\lim_{k\rightarrow\infty}\mu_{k}\). The quantum ergodicity theorem says that, for almost all sequences \(\mu_{k}\) the limit measure \(\mu\) is the Liouville measure. If this is true for all sequences, one speaks instead of quantum unique ergodicity (QUE), something which has only been proven in very special cases, and does in fact not hold in general. One example where QUE does not hold are quantized hyperbolic maps for the \(2\)-torus.
In the paper under review, the author wants to study certain ergodic properties of arbitrary limit measures \(\mu\). In particular, he proves a bound on the metric (Kolmogorov-Sinai) entropy \(H_{KS}(\mu)\) for a class of one-dimensional maps.
For the Laplacian on a compact Riemann surface \(X\) with Anosov geodesic flow on the unit cotangent bundle \(S^*X\) it was shown by N. Anantharaman and S. Nonnenmacher (cf., e.g., [Ann. Math. (2) 168, No. 2, 435–475 (2008; Zbl 1175.35036); Ann. Inst. Fourier 57, No. 7, 2465–2523 (2007; Zbl 1145.81033)]) that
\[ H_{KS}(\mu)\geq\int_{S^*X}|\log J^{u}(x)|\,d\mu-\tfrac{1}{2}\lambda_{\max}, \]
where \(J^{u}\) denotes the unstable Jacobian and \(\lambda_{\max}\) the maximum expansion rate of the flow. In some cases, the right hand-side might even be negative, and it therefore makes sense to try and prove a stronger inequality. It was conjectured by N. Anantharaman and S. Nonnenmacher [Ann. Henri Poincaré 8, No. 1, 37–74 (2007; Zbl 1109.81035); 2008, loc. cit.] that (under some conditions) a bound of the form
\[ H_{KS}(\mu)\geq\tfrac{1}{2}\int_{S^*X}|\log J^u(x)|\,d\mu \]
should hold.
The main result of the present paper is to prove that the analogue of this bound holds for a class of piecewise linear interval maps \(T_{p}:I\rightarrow I\), where \(I=\left[0,1\right]\) and the expansion rates are all of the form \(\Lambda_{j}=p^{n_{j}}\) for some fixed integer \(p>1\) and integers \(n_{j}\geq1\) (not necessarily equal). Examples are also given which show that the obtained bound is sharp (in certain cases).
The technique of the proof is to construct a uniformly expanding “tower” \(\tilde{T}:\tilde{I}\rightarrow\tilde{I}\) corresponding to \(T:I\rightarrow I\). The tower construction commutes with the quantization (at least of the “tensorial” type considered) and there is a simple relationship between the invariant measures as well as the metric entropies. By proving the bound for the tower (which is simpler since it is uniformly expanding) the corresponding result for the original system follows immediately.
As an illustration of the method, the author makes all arguments in the proof explicit for the very simplest non-uniform case, namely \(T_{\{2,4,4\}}:I\rightarrow I\) with partition \(I_{1}= [0,\frac{1}{2}],\) \(I_{2}=[\frac{1}{2},\frac{3}{4}]\), \(I_{3}=[\frac{3}{4},1]\) and expansion rates \(\Lambda_{1}=2\), \(\Lambda_{2}=\Lambda_{3}=4\). Following this example throughout makes the proof easier to understand.
The author also states that the desired bound can in fact be proven for all one-dimensional maps whose slopes are given by integer powers of integers (and general quantization).