Truong’s l-adic cohomology version of the dynamical degree–cohomology spectral radius equality in positive characteristic

Establish, in positive characteristic, the equality between dynamical degrees and the spectral radii of pullbacks on l-adic cohomology: for every smooth projective variety X over an algebraically closed field of positive characteristic and every dominant endomorphism f: X → X, prove that for each i the i-th dynamical degree λ_i(f) equals the spectral radius of the induced map f*: H^{2i}(X, Q_l) → H^{2i}(X, Q_l) for every prime l different from the characteristic of the base field.

Background

In characteristic zero, Dinh proved that for smooth projective varieties and surjective endomorphisms the i-th dynamical degree equals the spectral radius of the induced pullback on singular cohomology, establishing a deep cohomological interpretation of dynamical degrees.

Truong proposed extending this equality to positive characteristic by replacing singular cohomology with l-adic cohomology. The conjecture is significant and challenging: even the special case of Frobenius endomorphisms would imply Deligne’s proof of the Weil conjectures.

Partial progress is known: Esnault–Srinivas proved related equalities for surface automorphisms, and Truong proved a maximal spectral radius equality for all degrees; however, the full equality for each i remains open.