Estimates of Hausdorff dimension for non-wandering sets of higher dimensional open billiards (Q2862946)

The author investigates open billiards with domain \(Q=\mathbb{R}^D\setminus K\), \(D\geq 2\), where \(K\) decomposes into a disjoint union \(K=K_1\cup\ldots\cup K_n\), \(n\geq 3\), of compact, strictly convex sets (the obstacles) with smooth boundaries. To ensure that the non-wandering set \(M_0\) of the considered billiard does not contain trajectories tangent to the boundary, he assumes further that the obstacles are non-eclipsing, i.e., for each three disjoint obstacles \(K_i, K_j, K_k\), the convex hull of \(K_i\cup K_j\) does not intersect \(K_k\). Let \(d_{\text{min}}\) respectively \(d_{\text{max}}\) denote the minimal respectively maximal distance between points on distinct obstacles seen by the non-wandering set \(M_0\). Set NEWLINE\[NEWLINE \lambda = (1 + d_{{\max}} g_{{\max}})^{-1} \quad\text{and}\quad \mu = (1+d_{{\min}} g_{{\min}})^{-1}, NEWLINE\]NEWLINE where \(g_{\text{max}}\) and \(g_{{\min}}\) are certain constants depending on the curvature at the convex front. The main result of this paper are the following estimates of the Hausdorff dimension of \(M_0\).NEWLINENEWLINE{ Theorem:}NEWLINENEWLINE(i) For \(D=2\) we have NEWLINE\[NEWLINE - \frac{2\ln(n-1)}{\ln \lambda} \leq \text{dim}_H M_0 \leq - \frac{2\ln(n-1)}{\ln \mu}. NEWLINE\]NEWLINENEWLINENEWLINE(ii) Let \(\alpha = \frac{2 d_{{\min}}\ln \mu}{d_{{\max}} \ln \lambda}\). For any \(D\geq 2\) we have NEWLINE\[NEWLINE -\alpha \frac{2\ln(n-1)}{\ln \lambda} \leq \text{dim}_H M_0 \leq -\frac{1}{\alpha} \frac{2\ln(n-1)}{\ln\mu}. NEWLINE\]NEWLINENEWLINENEWLINE(iii) If the obstacles \(K_i\) are so far apart such that \(\lambda^{d_{{\max}}} < \mu^{2d_{{\min}}}\), then the estimate (i) also holds for \(D\geq 3\). NEWLINENEWLINENEWLINEIn addition the author proposes a conjecture on the structure of periodic billiard trajectories that confines the non-wandering set, which he proves for the case of a \(3\)-dimensional billiard with spheres as obstacles. Assuming this conjecture, the requirements of the main theorem can be relaxed.