Finitude of physical measures for Markovian random maps

The paper proves Theorem E by reducing the Markov-driven system to an i.i.d. representation h, showing h is Lipschitz with an exponential moment, applying a Bernoulli (i.i.d.) mostly-contracting result to obtain quasi-compactness of the annealed Koopman operator, and then transferring both quasi-compactness and (fpm) back to the original Markov setting via the conjugacy π and the characterization of P-Markovian invariant measures (notably Theorem 3.2 and the basin-transfer lemmas) . By contrast, the candidate solution hinges on deducing Λ=∑t π(t)log Lip(ft)<0 from the assumption λ(f)<0 (sup over P-Markovian invariant measures of maximal Lyapunov exponents), and then perturbing ρ(Q Dα). But λ(f)<0 only implies λ(μ̄)≤Λ for every P‑Markovian μ̄; it does not force Λ<0. Hence the key spectral step ρ(Q Dα)<1 is unjustified under the paper’s hypotheses. The candidate further asserts uniqueness of a physical measure (r=1) via a contraction in a Hölder dual metric, which is strictly stronger than (and not implied by) the paper’s (fpm) conclusion of finiteness; the paper never claims uniqueness, and in general only finiteness can be guaranteed .