论文标题
Newhouse厚度定理的非线性版本
A nonlinear version of the Newhouse thickness theorem
论文作者
论文摘要
令$ C_1 $和$ C_2 $是两个cantor套装，带有凸壳$ [0,1] $。 Newhouse证明，如果$τ（C_1）\ CDOTτ（C_2）\ geq 1 $，则算术和$ C_1+C_2 $是一个间隔，其中$τ（C_I），1 \ leq I \ leq 2 $ 2 $表示$ C_I $的厚度。在本文中，我们将此厚度定理概括如下。令$ k_i \ subset \ mathbb {r}，i = 1，\ cdots，d $，是一些cantor set（完美且无处可去），带有convex hull $ [0,1] $。假设$ f（x_1，\ cdots，x_ {d-1}，z）\ in \ mathcal {c}^1 $是在$ \ mathbb {r}^d $上定义的连续函数。表示$ f $ by $$ f（k_1，\ cdots，k_d）的连续图像= \ {f（x_1，\ cdots x__ {d-1}，z）：x_i \ in k_i，z \ in k_d in k_d，1 \ leq i \ leq i \ leq i \ leq d-1 \}。 x_ {d-1}，z）\ in [0,1]^d $，我们有$$（τ（k_i））^{ - 1} \ leq \ left | \ dfrac {\ partial_ {x_i} f} f} f} $ f（k_1，\ cdots，k_d）$是封闭的间隔。我们提供两个申请。首先，我们部分回答了高桥提出的一些问题。其次，我们获得了各种非线性身份，这些身份与持续的局部局部数量相关的分数相关，可以代表实数。
Let $C_1$ and $C_2$ be two Cantor sets with convex hull $[0,1]$. Newhouse proved if $τ(C_1)\cdot τ(C_2)\geq 1$, then the arithmetic sum $C_1+C_2$ is an interval, where $τ(C_i), 1\leq i\leq 2$ denotes the thickness of $C_i$. In this paper, we generalize this thickness theorem as follows. Let $K_i\subset \mathbb{R}, i=1,\cdots, d$, be some Cantor sets (perfect and nowhere dense) with convex hull $[0,1]$. Suppose $f(x_1,\cdots, x_{d-1},z)\in \mathcal{C}^1$ is a continuous function defined on $\mathbb{R}^d$. Denote the continuous image of $f$ by $$f(K_1,\cdots, K_d)=\{f(x_1, \cdots x_{d-1},z):x_i\in K_i,z\in K_d, 1\leq i\leq d-1\}.$$ If for any $(x_1, \cdots, x_{d-1},z)\in [0,1]^d$, we have $$(τ(K_i))^{-1}\leq \left|\dfrac{\partial_{x_i} f}{\partial_z f}\right|\leq τ(K_d),1\leq i\leq d-1$$ then $f(K_1,\cdots, K_d)$ is a closed interval. We give two applications. Firstly, we partially answer some questions posed by Takahashi. Secondly, we obtain various nonlinear identities, associated with the continued fractions with restricted partial quotients, which can represent real numbers.
