Almost all orbits of an analogue of the Collatz map on the reals attain bounded values

Motivated by a balanced ternary representation of the Collatz map we define the map $C_{\mathbb{R}}$ on the positive real numbers by setting $C_{\mathbb{R}}(x)=\frac{1}{2}x$ if $[x]$ is even and $C_{\mathbb{R}}(x)=\frac{3}{2}x$ if $[x]$ is odd, where $[x]$ is defined by $[x]\in \mathbb{Z}$ and $x-[x]\in (-\frac{1}{2},\frac{1}{2}]$. We show that there exists a constant $K>0$ such that the set of $x$ fulfilling ${\mathrm{lim\hspace{0.17em}inf}}_{n\in \mathbb{N}}{C}_{\mathbb{R}}^n(x)\le K$ is Lebesgue-co-null. We also show that for any $\epsilon >0$ the set of $x$ for which $(\frac{3^{\frac{1}{2}}}{2}{)}^k{x}^{1-\epsilon }\le C_{\mathbb{R}}^k(x)\le (\frac{3^{\frac{1}{2}}}{2}{)}^k{x}^{1+\epsilon }$ for all $0\le k\le \frac{1}{1-\frac{{\log }_{2}3}{2}}{\log }_{2}x$ is large for a suitable notion of largeness.