Let be a closed manifold (mostly 3 dimension) and a foliation on . A leaf is closed if (like a periodic orbit or a genus- surface). It is proper if is closed (like the graph of ). It is recurrent if it is either closed (trivial proper) or non-proper (like a line on with irrational slope). Let be the part of closed leaves (note that may not be closed, like the periodic orbits of Anosov flow). Let be the part of proper leaves (nontrivial proper: not closed) . In particular all leaves outside are recurrent.
The foliation is said to be non-wandering if has no interior. And is said to be recurrent if , that is, every leaf of is recurrent. Moreover, is said to be almost periodic, if forms a new decomposition of (either disjoint or coincide). Yokoyama observed the following proposition:
Proposition. A almost periodic foliation is recurrent.
Proof. Let be a non-closed element and . Clearly . Then almost periodicity implies that . So . So every non-closed leaf is not proper and is recurrent.
A foliation is said to be R-closed if is a closed subset.
2. A support function on a convex domain is the signed distance. Let be a closed strictly convex domain around the origin. Then its support function is given by . It is easy to see that the supreme is attained at a point on the boundary, whose oriented tangent line has angle with positive -axis. Using this parameter the billiard system admit a coordinate , where is the angle of the out-going vector with the tangent direction. In particular .
Let be a closed piecewise-smooth convex curve around the origin, $\phi$ the angle of the tangent line at a point with -axis (serving as a parametrization, so and the line ). Then the distance from to the tangent line is .
Taking derivative with respect to , we get .
1. Phase transitions in statistical mechanics. A phase transition occurs when a material changes its properties in a dramatic way. For example water, as it is cooled and turns into ice. Phase transitions are characterized by an order quantity (like density) that changes as a function of a parameter of the system (such as the temperature). The special value of the parameter at which the system changes its phase is the system’s critical point.
A bifurcation occurs in a dynamical systems, when a small/smooth change of the parameter values (the bifurcation parameters) of a system causes a sudden ‘qualitative’ or topological change in its behavior. For example the ‘period-doubling bifurcation’ of Logistic map, the saddle-node bifurcation.
Phase transition in dynamical systems
– the parameters $t$’s where the pressure $P(f,t\phi)$ fail to be $C^k$ for $k=0,1,\cdots,\infty,\omega$;
– the parameters $t$’s where the system $f_t:M^2\to M^2$ shifts from integrability to nonintegrability, from regular to chaotic.