Positive entropy invariant measures on the space of lattices with escape of mass (Q2884078)

The author considers the space of modular lattices to construct a sequence of measures which weakly converges to the zero measure, while the limit of the sequence of entropies of the measures is not zero. This construction is motivated by the problem of estimating the mass escape of limit measures by controlling the entropy. The setting of the work is the following: Let \(X = \mathrm{SL}_{d+1} (\mathbb{Z}) \mathrm{SL}_{d+1} (\mathbb{R})\), \(d \geq 1\). This homogenous space can be identified with the space of unimodular lattices in \(\mathbb{R}^d\). Let \(T : X \rightarrow X\) be the transformation defined by \(T (x) = ax\), with \(a = \mathrm{diag}(e^{1/d}, e^{1/d}, \dots , e^{- 1} ) \in \mathrm{SL}_{d+1} (\mathbb{R})\). The main result is the following:NEWLINENEWLINE There exists a sequence of \(T\) invariant measures \((\mu_n)\) whose \({*}\)- limit is the zero measure and for which \(\lim_{n \to \infty} h_{{ \mu}_{n}} (T ) = d + 1\).NEWLINENEWLINE A corollary of this theorem is the following: For any \(c \in [0, 1]\) there is a sequence of measures \(({\mu}_{n})\) with \({*}\)-limit \(\mu \) such that \(h_ {{ \mu}_{n}}(T ) \geq c\) and \(\mu(X) \geq c -d\).NEWLINENEWLINEThe proof of the main theorem is given following these steps:NEWLINENEWLINE Let \(M > 0\) and \(x \in X\), and denote by \(ht(x)\) the inverse of the length of the shortest vector in \(x\). Define NEWLINE\[NEWLINEX_{<M} = \{x : ht(x) < M\}\quad \text{and}\quad X_{\geq M} = \{x : ht(x) \geq M\}.NEWLINE\]NEWLINE It is proved that there is a number \(M_0\) and a set set \(S_N \subset X_{<M}\) such that \(T^n (x) \in X_{{<M}_{0}}\) for any \(x \in S_N\). Once is established that there are infinitely many points in \(X_{<M}\) whose forforward orbits have heights \(ht\) above \(M\), a measure \(\mu\) can be constructed such that \(h_{\mu} (T ) > d -\varepsilon\) and \(\mu (X_{\geq M}) >1 -\varepsilon\). From this result the main theorem is deduced, since if for a fixed natural \(n\) one considers a measure \(\mu_n\) with \(h_{\mu_{n}} (T ) > d -1/n\) and \(\mu (X_{\geq n}) >1 - 1/n\) then the sequence \((\mu_n)\) \({*}\)-converges to a measure with mass \(0\).