Invariance of the symbolic dynamics under the complex symbolic space

Let $f$ be the rational function considered in the paper. Let $\mathcal{A}$ be the alphabet whose symbols $I_i$ correspond to the intervals produced by $pa$ on the real axis, and let $\Sigma_c=\mathcal{A}^{\mathbb{N}}$ be the associated classical symbolic space. Let $\mathcal{B}=\{J_1,\ldots,J_{14}\}$ and let $\Sigma=\mathcal{B}^{\mathbb{N}}$ be the symbolic space obtained using the map $spa:\mathbb{R\cup\{\infty\}}\longrightarrow\mathcal{B}$. Symbolic-dynamics invariance conjecture. The symbolic dynamics of $f$ does not change if one uses $\Sigma$ instead of $\Sigma_c$. This asserts that replacing the classical interval-based symbolic space by the symbolic space based on the $J_i$ symbols preserves the dynamical information, despite the latter construction incorporating the compactified real line. The supplied text does not state whether this claim has been proved or disproved.