Let $\mathbf{ Y}$ be the solution space of an $n$-layer cellular neural network, and let $\mathbf{ Y}^{ (i)}$ and $\mathbf{ Y}^{ (j)}$ be the hidden spaces, where $1 \leq i, j \leq n$. ($\mathbf{ Y}^{ (n)}$ is called the output space.) The classification and the existence of factor maps between two hidden spaces, that reaches the same topological entropies, are investigated in [Ban et al., J.~Differential Equations \textbf{ 252}, 4563-4597, 2012]. This paper elucidates the existence of factor maps between those hidden spaces carrying distinct topological entropies. For either case, the Hausdorff dimension $\dim \mathbf{ Y}^{ (i)}$ and $\dim \mathbf{ Y}^{ (j)}$ can be calculated. Furthermore, the dimension of $\mathbf{ Y}^{ (i)}$ and $\mathbf{ Y}^{ (j)}$ are related upon the factor map between them.
Categories: math.DS