A dynamical approach to Frobenius splitting

A dynamical system can be associated to the process of decomposing iterated Frobenius pushforwards of modules into indecomposable direct summands. Via this connection, we apply the theory of Markov chains to study asymptotic counts of multiplicities of indecomposable summands in the iterated Frobenius pushforwards of the given module, as the number of iterations grows. Our results recover and extend the existing results on the existence of the (generalized) F-signature due to Hashimoto--Symonds, Hashimoto--Kobayashi, Smith--Van den Bergh, Tucker, and Yao. We also develop further the theory of F-contributors, the modules for which the multiplicity grows at the maximal possible rate. As a key algebraic input, we define and develop strongly F-regular modules, extending Hochster--Huneke's notion of strongly F-regular rings. We use strongly F-regular modules to prove convergence of the relevant sequences and algebraically characterize F-contributors, when the underlying ring admits a module-finite strongly F-regular extension or has dimension at most two.