PowerIteration

The power iteration method is a simple iterative method for finding the eigenvector associated with the largest eigenvalue of a linear operator, such as the transfer operator of a dynamical system. The method works by repeatedly applying the operator to an initial guess for the eigenvector and normalizing the result to have unit length. The sequence of vectors obtained in this way converges to the eigenvector associated with the largest eigenvalue, under certain conditions.

To apply the power iteration method to the transfer operator associated with a dynamical system, we start with an initial guess for the invariant measure, which can be any non-negative function that integrates to $1$. We then apply the transfer operator to this function, and normalize the result to have unit integral. This gives us a new estimate for the invariant measure. We repeat this process, applying the transfer operator to the new estimate and normalizing the result, until the sequence of estimates converges to the invariant measure.

Mathematically, the power iteration method can be expressed as follows. Let $L$ be the transfer operator associated with the dynamical system, and let $\mu_0$ be the initial guess for the invariant measure. We then compute a sequence of functions $\mu_1, \mu_2, \mu_3, \dots$ using the formula:

$$\mu_{n+1}(x) = \frac{L[\mu_n(x)]}{\int L[\mu_n(x)] dx}$$

where the integral is taken over the domain of the dynamical system, and the normalization ensures that the resulting function has unit integral. We continue this process until the sequence ${\mu_n}$ converges to a fixed point, i.e., when $\mu_{n+1} = \mu_n$ for some value of $n$. This fixed point is the invariant measure associated with the transfer operator $L$.

The convergence of the power iteration method depends on the spectral properties of the transfer operator. In particular, if the transfer operator has a unique dominant eigenvalue (i.e., an eigenvalue with the largest absolute value), then the power iteration method converges to the corresponding eigenvector, which is the invariant measure. If there are multiple dominant eigenvalues, the convergence may be slower or the method may not converge at all.