Obtaining an explicit interval for a nonlinear Newhouse thickness theorem

Abstract

Let $C_1$ and $C_2$ be two Cantor sets in ${\mathbb {R}}$. Suppose that the size of the largest gap of $C_1$ is not greater than the diameter of $C_2$, and vice versa. Newhouse (Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, volume 35 of Cambridge Studies in Advanced Mathematics. Fractal dimensions and infinitely many attractors. Cambridge University Press, Cambridge, 1993) proved if $\tau (C_1)\cdot \tau (C_2)\ge 1$, then the arithmetic sum $C_1+C_2$ is an interval, where $\tau (C_i), 1\le i\le 2$ denotes the thickness of $C_i$. In this paper, we generalize this thickness theorem as follows. Let $K_i\subset {\mathbb {R}}, i=1,\ldots , d$, be some Cantor sets in ${\mathbb {R}}$. Suppose $f(x_1,\ldots , x_{d-1},z)\in {\mathcal {C}}^1$ is defined on ${\mathbb {R}}^d$. Denote the continuous image of f by

$$\begin{aligned} f\left( K_1,\ldots , K_d\right) =\left\{ f\left( x_1, \ldots , x_{d-1},z\right) :x_i\in K_i,z\in K_d, 1\le i\le d-1\right\} . \end{aligned}$$

In this paper, we give a sufficient condition under which $f(K_1,\ldots , K_d)$ is a closed interval. Our idea can reprove the Newhouse thickness theorem. Various applications are given.