http://www.scholarpedia.org/article/Hyperbolic_dynamics#:~:text=Among%20smooth%20dynamical%20systems%2C%20hyperbolic%20dynamics%20is%20characterized,semilocal%20or%20even%20global%20information%20about%20the%20dynamics.
什么是双曲动力系统？
A hyperbolic dynamical system is a type of dynamical system that exhibits a particular kind of behavior known as hyperbolicity. This behavior is characterized by the presence of expanding and contracting directions in the tangent space at each point of the system. Here is a strict mathematical definition:
Let ( M ) be a smooth manifold and ( f: M \to M ) be a diffeomorphism. The diffeomorphism ( f ) is said to be uniformly hyperbolic or an Anosov diffeomorphism if for every ( x \in M ) there is a splitting of the tangent space ( T_xM = E_s(x) \oplus E_u(x) ) and there are constants ( C > 0 ) and ( \lambda \in (0,1) ) such that for every ( n \in \mathbb{N} ) one has:
- ( |Df^n(v)| \leq C \lambda^n |v| ) for ( v \in E_s(x) ) (the stable subspace),
- ( |Df^{-n}(v)| \leq C \lambda^n |v| ) for ( v \in E_u(x) ) (the unstable subspace).
The subspaces ( E_s(x) ) and ( E_u(x) ) are called the stable and unstable subspaces at ( x ), respectively. They are invariant under the differential of ( f ), meaning ( Df(E_s(x)) \subset E_s(f(x)) ) and ( Df(E_u(x)) \subset E_u(f(x)) ).
This definition can be extended to flows. A flow ( \phi_t: M \to M ) is said to be uniformly hyperbolic or an Anosov flow if for every ( x \in M ) there is a splitting of the tangent space ( T_xM = E_s(x) \oplus E_0(x) \oplus E_u(x) ), where ( E_0(x) = \langle \phi_t’(x) \rangle ) is the flow direction, and there are constants ( C > 0 ) and ( \lambda \in (0,1) ) such that for every ( t > 0 ) one has:
- ( |D\phi_t(v)| \leq C \lambda^t |v| ) for ( v \in E_s(x) ),
- ( |D\phi_{-t}(v)| \leq C \lambda^t |v| ) for ( v \in E_u(x) ).
In both cases, the constants ( C ) and ( \lambda ) are independent of the point ( x ). This uniformity is a key feature of hyperbolic dynamical systems.
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