The Frobenius-Perron operator on spaces of curves

Abstract:

Let $\tau :{R^2} \to {R^2}$ be a diffeomorphism which leaves a compact set $A$ invariant. Let $B \subset A$ be such that $\tau$ can map out of $B$. Assume that $\tau$ has a hyperbolic fixed point $p$ in $B$. Let $\mathcal {C}$ be a space of smooth curves in $B$. We define a normalized Frobenius-Perron operator on the vector bundle of Lipschitz continuous functions labelled by the curves in $\mathcal {C}$, and use it to prove the existence of a unique, smooth conditionally invariant measure $\mu$ on a segment ${V^u}$ of the unstable manifold of $p$. A formula for the computation of ${f^{\ast }}$, the density of $\mu$, is derived, and $\mu ({\tau ^{ - 1}}{V^u})$ is shown to be equal to the reciprocal of the maximal modulus eigenvalue of the Jacobian of $\tau$ at $p$.