论文标题
强大的Borel-cantelli引理复发
A strong Borel--Cantelli lemma for recurrence
论文作者
论文摘要
考虑一个混合动力学系统$（[0,1]，t，μ）$，例如用Gibbs测量$μ$的分段扩展间隔图。给定一个非阴性数字的非符合序列$（m_k）$，可以定义$ r_k（x）$，使得$μ（b（x，r_k（x）= m_k $。证明，几乎所有$ x $ y几乎所有$ x $，$ k \ leq n $的$ t^k（x）$ t^k（x） M_n $。 结果是\ [ \ lim_ {r \ to 0} \ frac {\logτ_{b（x，r）}（x）} { - \logμ（b（x，x，r））} = 1 \]几乎每个$ x $，其中$τ$是返回时间。
Consider a mixing dynamical systems $([0,1], T, μ)$, for instance a piecewise expanding interval map with a Gibbs measure $μ$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $μ(B(x, r_k(x)) = m_k$. It is proved that for almost all $x$, the number of $k \leq n$ such that $T^k (x) \in B_k (x)$ is approximately equal to $m_1 + \ldots + m_n$. This is a sort of strong Borel--Cantelli lemma for recurrence. A consequence is that \[ \lim_{r \to 0} \frac{\log τ_{B(x,r)} (x)}{- \log μ(B (x,r))} = 1 \] for almost every $x$, where $τ$ is the return time.
