论文标题
一组间隔分段仿射收缩的Hausdorff尺寸
Hausdorff dimension of the exceptional set of interval piecewise affine contractions
论文作者
论文摘要
令$ i = [0,1）$，$ -1 <λ<1 $和$ f \ colon i \ to i $是一个分段$λ$ - $λ$ - 额外的$ i $的映射，即，存在一个分区$ 0 = a_0 = a_0 <a_1 <a_1 <a_1 <a_1 <\ cdots <a_ {k-1} $ nifters uniont unions uniont $ $ b_1，\ ldots，b_k \ in \ mathbb {r} $，以至于每一个$ f（x）=λx+ b_i $ in [a_ {i-1}，a_ {i-1}，a_ {i}），a_ {i}）$和$ i = 1，\ ldots，k $。 $ f $的特殊集合$ \ MATHCAL {e} _f $是\ Mathbb {r} $的一组参数$δ\，因此$r_Δ\ circ f $不是渐近的周期性，其中$r_δ\ colon i \ to i $ to i $ to i $ to i $ to the the Angle $Δ$ $δ$。在本文中，我们证明$ \ Mathcal {e} _f $的Hausdorff尺寸为零。我们从具有独立兴趣的$ \ mathbb {r} $上的分散lipschitz收缩的更一般定理中得出了这一结果。
Let $I=[0,1)$, $-1<λ<1$ and $f\colon I\to I$ be a piecewise $λ$-affine map of the interval $I$, i.e., there exist a partition $0=a_0<a_1<\cdots< a_{k-1}<a_k=1$ of the interval $I$ into $k\geq2$ subintervals and $b_1,\ldots, b_k\in\mathbb{R}$ such that $f(x)=λx+ b_i$ for every $x\in[a_{i-1},a_{i})$ and $i=1,\ldots,k$. The exceptional set $\mathcal{E}_f$ of $f$ is the set of parameters $δ\in\mathbb{R}$ such that $R_δ\circ f$ is not asymptotically periodic, where $R_δ\colon I\to I$ is the rotation of angle $δ$. In this paper we prove that $\mathcal{E}_f$ has zero Hausdorff dimension. We derive this result from a more general theorem concerning piecewise Lipschitz contractions on $\mathbb{R}$ that has independent interest.
