On the structure of multi-layer cellular neural networks: Complexity between two layers

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.