On two noteworthy deformations of negatively curved Riemannian metrics (Q1576838)

Let \(M\) be a closed connected \(C^\infty\) Riemannian manifold whose geodesic flow \(\varphi\) is Anosov. Let \(\theta\) be a smooth 1-form, which can be considered as a function \(\theta:T M\to\mathbb{R}\). For \(\lambda\in \mathbb{R}\) consider the 1-parameter family of convex superlinear Lagrangians \(L_\lambda: TM \to\mathbb{R}\) given by \[ L_\lambda(x,v) =\frac{1}{2} |v|^2_x-\lambda \theta_x (v) \tag{1} \] and let \(h_F(\lambda)\) be the topological entropy of the geodesic flow of the Finsler metric. The main goal of this note to compare various dynamical properties of two \(1\)-parameter deformations \(\varphi^\lambda_{EL}\) and \(\varphi^\lambda_F\) of the original geodesic flow. Here \(\varphi^\lambda_{E L}\) is the restriction of the flow generated by (1) to the unit sphere bundle. Define \(F_\lambda: TM\to\mathbb{R}\) by \(F_\lambda(x,v)= |v|_x-\lambda \theta_x (v)\). For sufficiently small \(\lambda\), \(F_\lambda\) defines a Finsler geodesic flow \(\varphi^\lambda_F\). The author shows that \[ h_EL''(0)+ h_F'' (0)=h^2 \text{Var} \theta, \tag{2} \] where \(\text{Var} \theta\) is the variance of \(\theta\) with respect to the measure of maximal entropy \(\varphi\).