We study a variant of branching Markov chains in which the branching is governed by a fixed deterministic tree $T$ rather than a Galton-Watson process. Sample path properties of these chains are determined by an interplay of the tree structure and the transition probabilities. For instance, there exists an infinite path in $T$ with a bounded trajectory iff the Hausdorff dimension of $T$ is greater than $\log(1/\rho)$ where $\rho$ is the spectral radius of the transition matrix.

Journal: Annals of Probability - ANN PROBAB, vol. 22, no. 1994, pp. 219-243, 1994

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