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Abstract: Probabilistic potential theory relates Markov chains and harmonic functions. It offers the tools to compute, for instance, the probability that a random walk hits a specified target before another; or, thinking in terms of open systems, that the walk leaves the space at a specified point.

We adapt those results to dynamical systems. Our aim is to estimate, in a spatially periodic hyperbolic system, the probability of hitting a specified target before another. This work involves tools from potential theory (specifically, a balayage identity), transfer operators with spectral degeneracies, and perturbative arguments.