Denjoy examples of class $C^1$ with affine dynamics outside the invariant Cantor set

What the math gives to ML

The paper gives a constructive Denjoy blow-up mechanism: an irrational circle rotation is lifted by replacing each orbit point with an interval of length lₙ, with Σₙ lₙ < ∞, producing an invariant zero-measure exceptional set and dynamics that are affine on every complementary interval. The transferable asset is an explicit semiconjugacy from simple neutral phase dynamics to piecewise-affine transport whose local expansion is controlled by adjacent interval lengths. A neural analogue is a phase-indexed recurrent memory whose cells transport hidden states with analytically specified gains. The sharp prediction is that the product of local transport slopes telescopes exactly to an endpoint-width ratio, allowing long-horizon memory distortion to be designed and monitored.