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Zero-area postcritical-set conjecture

For every c∈Cc\in\mathbb{C}, let fc(z)=z2+cf_c(z)=z^2+c and let P(fc)={fcn(0):n≥1}‾P(f_c)=\overline{\{f_c^n(0):n\ge 1\}} be its postcritical set. The conjecture asserts that m2(P(fc))=0m_2\bigl(P(f_c)\bigr)=0, where m2m_2 denotes planar Lebesgue measure.

References

Primary source

arXiv

Additional references

complex-dynamicsholomorphic-dynamics
Posted · edited

Progress summary

Refreshed
Claimed progress

A September 2026 paper reports that every neutral quadratic polynomial has a postcritical set of zero area, leaving the full conjecture open.

The conjecture asserts that every quadratic polynomial has a postcritical set of zero area. The latest result settles the entire neutral case, without arithmetic restrictions on the rotation number.

Known results

  • A 2024 preprint established zero area in several arithmetic cases, including Herman, Brjuno-but-not-Herman, and non-Brjuno rotation numbers, and proved an upper-semicontinuity theorem for postcritical sets.

September 2026 neutral-case result

Willie Rush Lim’s paper reports zero Lebesgue measure for postcritical sets of all neutral quadratic polynomials, removing the earlier arithmetic restrictions. This is a major unconditional special case, not a complete resolution of the conjecture.

Current status (as of September 2026): The neutral quadratic case is reported solved, while the full zero-area conjecture remains open.

Sources

Solutions 0

No solutions have been posted yet.