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?

  • $\begingroup$ Akhil -- out of curiosity: why do you call $f=A(g\circ T)-g$ a cohomological equation? $\endgroup$ – algori Jul 28 '10 at 22:42
  • $\begingroup$ I think it's called a "twisted" cohomological equation. The equation $f = g\circ T -g$ comes from group cohomology (it says that the cocycle generated by $f$ is a coboundary). $\endgroup$ – Akhil Mathew Jul 28 '10 at 22:55

Part I: The answer is yes under additional conditions:

  1. Periodic data conditions are satisfied. That is, for any periodic point $p$ $$ \sum_{x\in O(p)}f(x)=0. $$
  2. Exponent $\alpha$ is sufficiently close to 1.
  3. Transformation $A$ is dominated by $T$. That is, the map $(x,v)\mapsto(Tx, Av)$ is partially hyperbolic.

Then Walkden's paper "Solutions to the twisted cocycle equation over hyperbolic systems" proves that there exists an $\alpha$-Holder solution $g$. The result is more general: the target group is any Lie group with a bi-invariant metric and the equation is the cohomological equation for two cocycles rather than just coboundary equation.

Part II: Notice however that if $A\neq Id$ then the periodic conditions may be no longer necessary. Let's restrict to the case when $k=1$ then our equation takes form $$ f=\lambda g\circ T-g, $$ where $\lambda<1$. Direct computation shows that $$ g=-\sum_{i\ge 0} \lambda^if\circ T^i $$ is a solution. It is also clear that $g$ is Holder continuous. Moreover, in this case uniqueness is clear as well since the above formula for $g$ is obtained recurrently from $$ g=-f+\lambda g\circ T. $$ It seems that this generalizes rather straightforwardly to the case when $A$ is hyperbolic. And I think it's worthwhile to see if anything interesting happens in the case then $A$ has some eigenvalues on the unit circle and some off.

  • $\begingroup$ And of course periodic data conditions must be sutisfied! $\endgroup$ – Andrey Gogolev Jul 28 '10 at 22:54
  • $\begingroup$ See an update. $\endgroup$ – Andrey Gogolev Jul 29 '10 at 5:45

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