论文标题
在Lagrange和Markov Spectra的尺寸函数的图表上
On the graph of the dimension function of the Lagrange and Markov spectra
论文作者
论文摘要
我们研究了编码经典Lagrange和Markov Spectra的Hausdorff尺寸的函数$ d（t）$的图，并具有$（ - \ infty，t）$的半限制线。为此，我们使用这样一个事实，即在擦除其马尔可夫分区的元素以确定十二个非平凡的高原（t）$的元素后，动态cantor套件的Hausdorff尺寸下降。接下来，我们采用严格的数值方法（从最近与Pollicott的联合论文中）来产生这些十二个高原之间的$ d（t）$的近似值。作为推论，我们证明了$ d（t）$的最大的十个非平凡高原是那些长度> 0.005 $的高原。
We study the graph of the function $d(t)$ encoding the Hausdorff dimensions of the classical Lagrange and Markov spectra with half-infinite lines of the form $(-\infty, t)$. For this sake, we use the fact that the Hausdorff dimension of dynamically Cantor sets drop after erasing an element of its Markov partition to determine twelve nontrivial plateaux of $d(t)$. Next, we employ rigorous numerical methods (from our recent joint paper with Pollicott) to produce approximations of the graph of $d(t)$ between these twelve plateaux. As a corollary, we prove that the largest ten non-trivial plateaux of $d(t)$ are exactly those plateaux with lengths $> 0.005$.
