Entropy and escape of mass for Hilbert modular spaces (Q2905187)

In dynamical study of diagonal flows, one considers a sequence of probability measures invariant under a diagonal element of a linear group acting on a homogeneous space. It is known that the limit measure of the space could be any value in the unit interval \([0, 1]\). In this paper under review, the author studies the relation between metric entropy and escape of mass for Hilbert modular spaces with the action of a diagonal element.NEWLINENEWLINETo state the result, let \(F\) be an algebraic number field and let \({\mathcal O}\) be its ring of integers. Let \(S^{\infty}= \{\sigma_1,\ldots, \sigma_{r+s}\}\) be its Archimedean places where \(\{\sigma_1,\ldots, \sigma_r\}\) are the real places and the rest are complex ones. Define \(G := \prod_{n=1}^r \mathrm{SL}_2\left({\mathbb R}\right) \times \prod_{m=1}^{s}\mathrm{SL}_2 \left({\mathbb C}\right)\) and \(\Gamma := \mathrm{SL}_2\left({\mathcal O}\right).\) Then, \(\Gamma\) is naturally embedded into \(G\) via \(\Delta(\gamma) = \left(\sigma_1(\gamma), \ldots, \sigma_{r+s}(\gamma)\right)\) for \(\gamma\in \Gamma.\) Set \(X := \Gamma \backslash G\). Let \(a\) be any fixed diagonal element of \(G\) given by \(a = \mathrm{diag}(e^{i\theta_1}e^{a_1/2}, e^{-i\theta_1}e^{-a_1/2})\times \cdots\times \mathrm{diag}(e^{i\theta_{r+s}}e^{a_{r+s}/2}, e^{-i\theta_{r+s}}e^{-a_{r+s}/2})\) with \(a_j\in {\mathbb R}\), \(\theta_j\in [0, 2 \pi]\) and \(\theta_1, \ldots, \theta_r = 0.\) Define the action \(T\) on \(X\) by \(T(x) = x\cdot a.\)NEWLINENEWLINEFor a positive constant \(M\), define \(X_{< M} = \{x\in X: \mathrm{ht}(x) < M\}\) where \(\mathrm{ht}(\cdot)\) is called the height funtion on \(X\) defined in the paper. Let \(h_{\max}(T)\) denote the maximal metric entropy of \(T\). Then, one of the main results can be stated as follows. For \(M > \max\{e^{3 h_{\max}(T)}, 100\}\) be given. Then there exists a continuous decreasing function \(\phi : {\mathbb R}^{+}\to {\mathbb R}\) with \(\lim_{M\to\infty} \phi(M) = 0\) such that \(\mu(X_{<M}) \geq 1 - 2\left[(h_{\max}(T) - h_\mu(T))/h_{\max}(T)\right] - \phi(M)\). In particular, for a sequence of \(T\)-invariant masures \(\mu_n\) with \(h_{\mu_n}(T)\geq h\) one has that any weak* limit \(\mu_\infty\)has at least \(2h/h_{\max}(T) - 1\) mass left.NEWLINENEWLINEThe author also studies limits of a sequence of probability measures arising by averaging an arbitrary measure under iterates of \(T\). Let \(D = \dim U^{+} \leq r + 2s\) where \(U^+\) is the subgroup of \(G\) consisting of element \(g\in G\) such that \(a^{-n} g a^{n} \to 1\) as \(n\to -\infty.\) Let \(\nu\) be a probability measure of dimension at least \(d\) in the unstable direction with respect to \(a\) (here we omit the definition of the notion of ``dimension in the unstable direction'') and let \(\mu_n = \sum_{j=0}^{n-1} T_{\ast}^j \nu /n\) where \(T_{\ast}^j \nu\) is the push-forward of \(\nu\) under \(T^j.\) Then the author shows that for the sequence of probability measure \((\mu_n)_{n\geq 1}\), any weak* limit \(\mu_\infty\) has at least \(\mu_\infty(X) \geq 1 - 2 a_{\ast}(D-d)/h_{\max}(T)\) mass left where \(a_{\ast} = \max\{|a_i| \mid i = 1,\ldots, r+s\}.\)