A note on combining chaotic dynamical systems using the fuzzy logic XOR operator.

The paper’s key step asserts that once f xor g is full-branch with bounded distortion, “the Lebesgue measure is ergodic,” and therefore the map is transitive and hence chaotic; however no invariance of Lebesgue measure is proved (full-branch + bounded distortion does not in general imply Lebesgue invariance), so the transitivity/chaos conclusion is unsupported . The candidate solution correctly reduces f xor g to φ∘f with φ(t)=|2t−1| and refines the partition to show h is full-branch—matching the paper’s Proposition 8 . But it then claims a “one-step hitting” property to deduce topological transitivity; that argument is invalid because h^{-1}(V) need not be dense for arbitrary open V, so transitivity is not established. In short: the paper’s proof of chaos is flawed (measure step), and the model’s proof of transitivity is flawed (topological step).