Denjoy counterexamples at the logarithmic derivative-modulus barrier
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Can an orientation-preserving C¹ circle diffeomorphism with irrational rotation number have a wandering interval when the derivative has logarithmic modulus of continuity ω(t)=t log(1/t)? This tests the regularity barrier for Denjoy counterexamples.
Why it matters and what counts as progress
Why it matters: The logarithmic modulus marks a boundary between classical regularity conditions and constructions with wandering intervals.
What counts as progress: Construct a wandering-interval example at the barrier or prove that a sharper regularity condition forces the Denjoy conclusion.
How the problem developed
- 1932 · Theorem [1]
Arnaud Denjoy Showed that a circle diffeomorphism with derivative of bounded variation has no wandering intervals; this is a regularity threshold behind the later modulus question.
- 1979 · Counterexample [1]
Michael Herman Constructed examples at moduli t log^{1+ε}(1/t), just above the t log(1/t) barrier.
- 3 Aug 2026 · Theorem [1]
Rohil Prasad The preprint constructs irrational circle diffeomorphisms with wandering intervals at every concave modulus weaker than Lipschitz, including the logarithmic barrier case.
AI contribution
On the sharpness of Denjoy's theorem
The retained AI-origin construction gives a C¹ circle diffeomorphism with irrational rotation number, a wandering interval and derivative modulus ω(t)=t log(1/t), the barrier case in Prasad’s Corollary 1.1.
Sources
Website timeline
Publication dates and editorial changes.
AI-origin construction reaches the Denjoy logarithmic barrier