Stable Hartman conjecture

Let $f$ be an analytic map defined near a fixed point $p$ , with $f(p)=p$ . Assume that the linearization $A=Df(p)$ is semisimple and fully stable, meaning that every eigenvalue of $A$ has modulus less than $1$ . The stable Hartman conjecture asserts that, for every $\varepsilon>0$ , there are neighborhoods of $p$ and $0$ and a local conjugacy $h$ of $f$ to its linear part such that $h(p)=0$ , $h\circ f\circ h^{-1}(x)=Ax$ , and $h$ has regularity $C^{1,1-\varepsilon}$ . More generally, in the presence of resonances, the stable invariant manifolds and linearizing maps are expected to have the corresponding regularity $C^{r,1-\varepsilon}$ determined by the minimal resonant index.

References

Primary source

arXiv

Additional references

Existence and Regularity of Stable Resonant Spectral Submanifolds and Linearization Maps — arXiv — Florian Kogelbauer, Rafael de la Llave

dynamical-systems normal-form-theory

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A new paper claims to prove the stable form of the conjecture under explicit assumptions, but broader versions remain unsettled.

The stable Hartman conjecture concerns linearizing analytic dynamical systems near stable behavior. The reported result derives stable invariant manifolds and linear conjugacies with regularity $C^{r,1-}$ under semisimplicity and stability hypotheses.

September 2026 claimed proof

Florian Kogelbauer and Rafael de la Llave report a logarithmic-polynomial functional expansion proving the stable Hartman conclusion under semisimple linearization. The claim is conditional on the stated analytic, spectral, and stability assumptions and has not been independently verified in the retrieved sources.

Current status (as of September 2026): A conditional proof of the stable conjecture is claimed but unverified; broader Hartman-type settings remain open.

Sources

No solutions have been posted yet.