In a Birkhoff region of instability for an exact area-preserving twist map, we construct some orbitsconnecting distinct Denjoy minimal sets. These sets correspond to the local, instead of global minimum of theLagrangi...In a Birkhoff region of instability for an exact area-preserving twist map, we construct some orbitsconnecting distinct Denjoy minimal sets. These sets correspond to the local, instead of global minimum of theLagrangian action. In the earlier work, Mather constructed connecting orbits among Aubry-Mather sets andthe global minimizer of the Lagrangian action.展开更多
Suppose that C 1 and C 2 are two simple curves joining 0 to ∞, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, the set {θ : re iθ∈ D} has measure at m...Suppose that C 1 and C 2 are two simple curves joining 0 to ∞, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, the set {θ : re iθ∈ D} has measure at most 2α, where 0 < α < π. Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = Φ(|z|) for all large z ∈ C 1 ∪ C 2 ∪～D, where Φ(|z|) is a convex, non-decreasing function of |z| and ～D is the complement of D. Let A D (r, u) = inf{u(z) : z ∈ D and |z| = r}. It is shown that if A D (r, u) = O(1) then lim inf r→∞ B(r, u)/r π/(2α) > 0.展开更多
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19525103).
文摘In a Birkhoff region of instability for an exact area-preserving twist map, we construct some orbitsconnecting distinct Denjoy minimal sets. These sets correspond to the local, instead of global minimum of theLagrangian action. In the earlier work, Mather constructed connecting orbits among Aubry-Mather sets andthe global minimizer of the Lagrangian action.
文摘Suppose that C 1 and C 2 are two simple curves joining 0 to ∞, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, the set {θ : re iθ∈ D} has measure at most 2α, where 0 < α < π. Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = Φ(|z|) for all large z ∈ C 1 ∪ C 2 ∪～D, where Φ(|z|) is a convex, non-decreasing function of |z| and ～D is the complement of D. Let A D (r, u) = inf{u(z) : z ∈ D and |z| = r}. It is shown that if A D (r, u) = O(1) then lim inf r→∞ B(r, u)/r π/(2α) > 0.
