Derivatives of topological entropy for Anosov and geodesic flows (Q1330390)

The author derives a simple formula for the first and second derivative of the topological entropy of a smooth perturbation \(\lambda \to \phi^{(\lambda)}\) of Anosov flows. Even more useful than the formula itself is the method used to obtain it: By structural stability such a perturbation can be realized as a family of suspensions over a fixed subshift \(\sigma: \Sigma \to \Sigma\) of finite type defined by a smooth family \(\lambda \to r^{(\lambda)}\) of Hölder continuous functions on \(\Sigma\). The entropy \(h^{(\lambda)}\) of \(\phi^{(\lambda)}\) is the unique zero of the assignment \(t \to \text{ pressure }(tr^{(\lambda)})\), and this is used to derive a formula for the second derivative of \(\lambda \to h^{(\lambda)}\) containing the variance of the first derivative \(D\alpha^{(\lambda)}\) of the velocity change \(\lambda \to \alpha^{(\lambda)}\) of \(\lambda \to \phi^{(\lambda)}\). In the case where \(\lambda \to \phi^{(\lambda)}\) is a family of geodesic flows of a smooth volume preserving family of metrics \(\lambda \to g^{(\lambda)}\) of negative curvature the function \(2D\alpha^{(\alpha)}\) is shown to coincide with the evaluation of the quadratic form defined by the first derivative of the metric tensors up to a coboundary. This gives rise to a purely geometric version of the formula for the second derivative. Finally estimates for the variance are given for conformal volume preserving perturbations of a hyperbolic metric on a compact 3-manifold.