Let $T: X \to X$ be an Anosov diffeomorphism. Suppose $f: X \to \mathbb{R}$ is Holder continuous (say with exponent $\alpha$). The question arises as to when $f$ can be written as $g \circ T - g $ for some $\alpha$-Holder $g: X \to \mathbb{R}$. It is easily checked that a necessary condition is that the periodic data vanishes, i.e. the sum of $f$ over every periodic orbit is zero. It is also a (nontrivial) theorem of Livsic that the converse is true, namely vanishing periodic data implies that this cohomological equation is solvable.

I'm interested in the following variant of the cohomological equation. Let $A: \mathbb{R}^k \to \mathbb{R}^k$ be an isomorphism. Suppose $f: X \to \mathbb{R}^k$ is $\alpha$-Holder. When is the equation $f = A( g\circ T) - g$ solvable in $g$?

There seem to be sufficient conditions known for the case of compact groups. Are there conditions known in the noncompact but abelian case?

Let $T: X \to X$ be an Anosov diffeomorphism. Suppose $f: X \to \mathbb{R}$ is Holder continuous (say with exponent $\alpha$). The question arises as to when $f$ can be written as $g \circ T - g $ for some $\alpha$-Hölder $g: X \to \mathbb{R}$. It is easily checked that a necessary condition is that the periodic data vanishes, i.e. the sum of $f$ over every periodic orbit is zero. It is also a (nontrivial) theorem of Livsic that the converse is true, namely vanishing periodic data implies that this cohomological equation is solvable.

I'm interested in the following variant of the cohomological equation. Let $A: \mathbb{R}^k \to \mathbb{R}^k$ be an isomorphism. Suppose $f: X \to \mathbb{R}^k$ is $\alpha$-Hölder. When is the equation $f = A( g\circ T) - g$ solvable in $g$?

There seem to be sufficient conditions known for the case of compact groups. Are there conditions known in the noncompact but abelian case?