Discrete Dynamical Systems with Random Impulses

The paper’s main theorem (Theorem 1.1) matches the model’s conclusion: under P^-(S)=1, T^n μ̂ converges weakly to the pushforward π̂ ⊙ P^- on M̂=N0×I (see the statement of Theorem 1.1 and the construction of P^- and π̂ in the paper ; the Markov operator is given in (5) ; the reversed kernel q and measure P^- are defined right before §1.3 ). The model’s proof is a pathwise synchronization/pullback argument using the two-sided stationary chain and time reversal, which differs from the paper’s measure-level approximation but yields the same limit. However, the paper contains a concrete algebraic mistake in simplifying the average-contraction threshold for the impulse case: from EP^-[log L_F0]=(1/(1+E))log L0 + (E/(1+E))log L1<0 the correct condition is L1 < L0^{-1/E}, not L1 < L0^{-E} as printed in (8); this reversal also flips the directions in Examples 5.2 and 5.4 (compare the derivation just above (8) and the statement of (8) and the examples in §5 ). Aside from this fix (and minor presentational issues), the paper’s core arguments and results align with the model’s solution.