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We show a Hopf type rigidity for thermostats without conjugate points on a 2-torus, 9 pages; minor revisions to reflect published version

We consider a general family of curves $\Gamma$ on a compact oriented Finsler surface $(M,F)$ with boundary $\partial M$. Let $\varphi\in C^{\infty}(M)$ and $\omega$ a smooth 1-form on $M$. We show that $$\int_{\gamma(t)}\{\varphi(\gamma(t))+\omega_{\gamma(t)}(\dot{\gamma}(t))\}\,dt=0$$ holds for every $\gamma\in\Gamma$ whose endpoints belong to $\partial M$, $\gamma(a)\in\partial M$, $\gamma(b)\in\partial M$ if and only if $\varphi=0$ and $\omega$ is exact. Similar results were proved when $M$ is closed and some additional conditions on Gaussian curvature are imposed. We also study the cohomological equations of Anosov generelized thermostats on a closed Finsler surface. Finally, we gave conditions when thermostat is of Anosov type., Comment: 25 pages

We consider a magnetic flow without conjugate points on a closed manifold $M$ with generating vector field $\G$. Let $h\in C^{\infty}(M)$ and let $\theta$ be a smooth 1-form on $M$. We show that the cohomological equation \[\G(u)=h\circ \pi+\theta\] has a solution $u\in C^{\infty}(SM)$ only if $h=0$ and $\theta$ is closed. This result was proved in \cite{DP2} under the assumption that the flow of $\G$ is Anosov.

We show that an arbitrary Anosov Gaussian thermostat on a surface is dissipative unless the external field has a global potential.

Let $M$ be a closed oriented surface and let $\Omega$ be a non-exact 2-form. Suppose that the magnetic flow $\phi$ of the pair $(g,\Omega)$ is Anosov. We show that the longitudinal KAM-cocycle of $\phi$ is a coboundary if and only the Gaussian curvature is constant and $\Omega$ is a constant multiple of the area form thus extending the results in \cite{P2}. We also show infinitesimal rigidity of the action spectrum of $\phi$ with respect to variations of $\Omega$. Both results are obtained by showing that if $G:M\to\mathbb R$ is any smooth function and $\omega$ is any smooth 1-form on $M$ such that $G(x)+\omega_{x}(v)$ integrates to zero along any closed orbit of $\phi$, then $G$ must be identically zero and $\omega$ must be exact.