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A. N. Livšic authored the 1972 paper Cohomology of dynamical systems, associated with the Livšic theorem. The original publication record and later primary research references verify these initials; they are retained without inferring a fuller personal identity.^[ref][ref]

For an equation $u\circ T-u=f$, summing along a periodic orbit telescopes to zero. Thus periodic sums of $f$ must vanish. Livšic theory shows that, in suitable hyperbolic settings with the required regularity assumptions, this necessary periodic-orbit test also yields a transfer function. The result turns infinitely many local orbit checks into a global solution of a cohomological equation.^[ref]

The Livšic theorem turns periodic-orbit data into a global derivative along a hyperbolic flow. Let ${\Phi }_t$ be a transitive Anosov flow on a compact smooth manifold, with generating vector field $F$, and let $h$ be a real Hölder continuous observable. There is a Hölder continuous function $u$, differentiable along flow lines, satisfyingif and only if ${\int }_0^{T}h({\Phi }_{t}p)dt=0$ for every periodic orbit $p$ of period $T$. The solution preserves the Hölder exponent.^[ref]

Necessity is immediate by putting ${\Phi }_{T}p=p$ in the identity. Two solutions differ by a continuous invariant function. Transitivity supplies a dense orbit; their difference is constant on that orbit, and continuity makes it constant everywhere. Thus the solution is unique up to an additive constant.

The harder sufficiency argument defines $u({\Phi }_t{x}_0)$ by integrating $h$ along a dense orbit. The Anosov closing lemma converts a sufficiently close return into a nearby periodic trajectory. Its vanishing integral and exponential shadowing bound the discrepancy between the endpoint values in terms of their distance. This gives Hölder control on the dense orbit and permits extension to the whole manifold.

When both the flow and $h$ are smooth, a separate regularity theorem gives smooth $u$. This does not follow merely from the preceding continuity argument.^[ref] The theorem is named for A. N. Livšic.

A periodic orbit obstruction is a necessary compatibility condition for solving a cohomological equation. If an dynamical observable is a flow coboundary, $h=Xu$, where $X$ differentiates along the flow, then on a periodic orbit of period $T$,A nonzero integral therefore rules out a single-valued transfer function $u$. Under suitable transitive Anosov flow and Hölder regularity hypotheses, the Livšic theorem provides a converse. Without those dynamical and regularity assumptions, vanishing periodic integrals alone is not a general existence theorem.^[ref]

Richard Sharp works on the geometry of Anosov flows. His Abelian Livšic theorems for Anosov flows by Richard Sharp gives two short proofs of an earlier abelian Livšic theorem and extends the framework to positive-density sets of null-homologous periodic orbits and amenable covers.^[ref]

The abelian formulation separates a derivative along the flow from a contribution carried by a closed differential form. Integrals over periodic orbits record the obstruction to representing a function entirely as such a derivative. This makes orbit data into information about cohomology and explains why restricting to null-homologous orbits changes the formulation.^[ref]