It is often said that Lyapunov exponents of Hamiltonian systems always come in conjugate pairs $\pm\lambda$. However, there are certainly examples that one can construct where Lyapunov exponents do not come in opposite pairs because time evolution makes vectors become collinear. An example would be the following time evolution in the vector space $V=\text{span}(x,p)$: \begin{align} \xi_1(t)&=p\,\\ \xi_2(t)&=x+e^t p\,. \end{align} Here, $p$ and $x$ are basis vectors that are conjugate through the symplectic form $\omega$ in the sense of $\omega(x,p)=1$. In this case, the initial point $\xi_1=p$ has Lyapunov exponent $\lambda_1=0$, while the initial points $\xi_2=x+p$ has Lyapunov exponent $\lambda_2=1$. However, time evolution is still Hamiltonian because the volume is preserved as $\xi_2$ is stretched exponentially in the direction of $\xi_1$ which means the vectors become collinear. In particular, the volume growth is not given by $\Lambda=\lambda_1+\lambda_2=1$, but it is just zero as it should be for Hamiltonian systems.
I'm not an expert in dynamical systems, but started to study them recently.
My questions:
- Is there a standard criterion to exclude weird cases as the one I just presented. Some class of "regular Hamiltonian systems" where time evolution does not let vectors approach collinearity exponentially fast.
- Can you recommend any compact references that discuss this subtlety of symplectic systems?
[Edit:] Let me add another question:
- I found several papers using the Oseledets theorem (multiplicative ergodic theorem) to show that Lyapunov exponents of Hamiltonian systems always come in opposite pairs. However, my example from above does not satisfy this. I assume that this is because one can interpret my example as a linear system on a linear phase space, such that the evolution of perturbations is the same everywhere. Therefore, time-averaging is something completely different than space-averaging. Is it true that linear systems on a linear space cannot be ergodic?
It is usually done differently: there are conditions that ensure some particular behavior of the Lyapunov exponents. For example, one often assume that the derivative (which appears in the linear variational equation) takes values in some particular group of matrices, which may lead or not to symmetric Lyapunov exponents.
As a starting point, I recommend:
the paper of M. Wojtkowski, Monotonicity, J-algebra of Potapov and Lyapunov exponents, in Smooth Ergodic Theory and its Applications, Amer. Math. Soc., 2001, pp. 499-521.
the book by Barreira and Pesin, Nonuniform Hyperbolicity, Cambridge University Press, Cambridge, 2007.