Minimal Measures for Lagrangian Systems

Fang Wang

doi:https://doi.org/10.21985/N25X6N

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Minimal Measures for Lagrangian Systems

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In this thesis we study minimal measures for Lagrangian systems on compact manifolds. This thesis consists of three parts which are closely related. The first part is Chapter 3 and Chapter 4. In Chapter 3 and 4, we consider geodesic flows on compact surfaces with higher genus. We show that for every rational vector $h\in H_1(M,\R)$ there is a minimal measures with rotation vector $h$ supported on a finite set of simple closed geodesics. We also prove that if a non-trivial simple closed geodesic has minimal arclength among all non-trivial simple closed curves, then the invariant probability measure evenly distributed on it is a minimal ergodic measure. The second part is Chapter 5, in which we study the intersection property of trajectories in supports of minimal measures for autonomous Lagrangian systems on surfaces. We show that for each pair of minimal measures $\mu_1$ and $\mu_2$, if the intersection number of $\rho(\mu_1)$ and $\rho(\mu_2)$ are non-zero, then any non-trivial convex combination of $\mu_1$ and $\mu_2$ is not a minimal measure because its support (which is the union of supports of $\mu_1$ and $\mu_2$) does not satisfy Mather's Lipschitz graph property. In Chapter 6 and Chapter 7, which is the third part of this thesis, we extend Mather's notion of minimal measures on manifolds with non-commutative fundamental groups and use finite coverings to study theses extended minimal measures. We study the existence of new minimal measures on finite-fold covering spaces in Chapter 6 for positive definite Lagrangian systems and show that, surfaces with higher genus has a richer set of minimal ergodic measures for geodesic flows. We define action-minimizers and minimal measures in the homotopical sense in Chapter 7 and prove the existence of minimal measures in the homotopical sense. We outline our programs in studying the structure of homotopical minimal measures by considering Mather's minimal measures on finite-fold covering spaces.

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