Let $X$ be a compact metric space. Given two covers $\mathcal{C}=(C_1, C_2, \ldots, C_n)$ and $\mathcal{D}=(D_1, D_2, \ldots, D_m)$ define their refinement as $\mathcal{C}\vee \mathcal{D}=\{C_i\cap D_j:i=1, 2, \ldots n, j=1, 2, \ldots m\}$.
Given a homeomorphism $f:X\to X$ and a cover $\mathcal{C}$, put $\mathcal{C}_0^n=\vee_{i=0}^n f^{-i}(\mathcal{C})$.
Call $f:X\to X$ scattering if for every finite cover $\mathcal{C}$ by non-dense open set has $r(\mathcal{C}_0^n)\to \infty$ as $n\to \infty$, where for finite open cover $\mathcal{D}$ of $X$, $r(\mathcal{D})$ is the minimal cardinality of a subcover of $\mathcal{D}$.
Is it true that $f:X\to X$ is scattering if for every finite cover $\mathcal{C}$ by non-dense open set, $r(\bigvee_{i\in\mathbb{N}} f^{-i}(\mathcal{C}))=\infty$?
Please help me to know it.