Given nonnegative \(N\times N\) matrices \({\mathbb Z}_1, \cdots, {\mathbb Z}_n\), the author introduces a stochastic process \(V({\mathbb Z}_1, \cdots, {\mathbb Z}_n) = \{V_j, j \geq 1\}\) [see the original article for the precise definition] and proves that, under the condition that \({\mathbb Z}_1+\cdots+ {\mathbb Z}_n\) is irreducible with its greatest eigenvalue being unity, \(V({\mathbb Z}_1, \cdots, {\mathbb Z}_n)\) is a hidden Markov chain which is stationary and ergodic.
The author provides a connection between the Hausdorff dimension of a self-similar measure (with overlaps) and the Shannon entropy of an associated hidden Markov chain. He demonstrates this connection by determining the Hausdorff dimension of the projection on the real line of the uniform measure on the Sierpinski gasket. Further development along this line can be found in the author’s recent article in [J. Math. Anal. Appl. 353, No. 1, 350–361 (2009; Zbl 1172.28003)].