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

Abstract

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 $1\le i,j\le n$. (${\mathbf{Y}}^{(n)}$ 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.

References in corpus (1)

• Measures of the full Hausdorff dimension for a general Sierpiński carpet