论文标题
随机$g_δ$ - 集的对数能力
Logarithmic capacity of random $G_δ$-sets
论文作者
论文摘要
我们研究间隔$ [0,1]的$g_Δ$子集的对数能力。$ s $ s $为f form \ begin \ begin {align*} s = \ bigCap_m \ bigCap_m \ bigCup_ {k \ ge m} i_k，\ end eend $ i_k $ a $ l_ a $ l_ $ 0 $。我们为$ s $提供足够的条件，即具有满负荷，即$ \ mathop {\ mathrm {cap}}（s）= \ mathop {\ mathrm {cap}}（[0,1]）$。当间隔呈指数衰减并将其放入$ [0,1] $相对于某些给定的分布中时，我们会考虑这种情况。这种分布产生的随机$g_δ$集几乎可以满足我们的足够条件，因此几乎可以肯定具有满载。这项研究是由$g_δ$集合在furstenberg定理的随机矩阵产品的参数版本中的特殊能量集。我们还研究了$g_δ$ sets $ \ {s（α）\} _ {α> 0} $，这些家庭通过将间隔的降低速度设置为$ l_k = e^{ - k^α}。
We study the logarithmic capacity of $G_δ$ subsets of the interval $[0,1].$ Let $S$ be of the form \begin{align*} S=\bigcap_m \bigcup_{k\ge m} I_k, \end{align*} where each $I_k$ is an interval in $[0,1]$ with length $l_k$ that decrease to $0$. We provide sufficient conditions for $S$ to have full capacity, i.e. $\mathop{\mathrm{Cap}}(S)=\mathop{\mathrm{Cap}}([0,1])$. We consider the case when the intervals decay exponentially and are placed in $[0,1]$ randomly with respect to some given distribution. The random $G_δ$ sets generated by such distribution satisfy our sufficient conditions almost surely and hence, have full capacity almost surely. This study is motivated by the $G_δ$ set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products. We also study the family of $G_δ$ sets $\{S(α)\}_{α>0}$ that are generated by setting the decreasing speed of the intervals to $l_k=e^{-k^α}.$ We observe a sharp transition from full capacity to zero capacity by varying $α>0$.
