An iterative map $T: x_{n+1} = T(x_n)$, given an initial state $x$, defines a periodic orbit:
$O_x=\lbrace x, S(x), ...,S^{n-1}(x) \rbrace$ ,
with $S^n(x) = x$. The periodic orbit supports a measure defined by
$\delta_{O_x} = \frac{1}{n} \sum_{i=0}^{n} \delta_{S^i(x)} $ ,
where $\delta_{S^i(x)})$ is the Dirac measure. This definition is given in ref. [1] pag. 38.
First of all, can we say that this measure corresponds to a uniform distribution?
Secondly, how can be written for continuous dynamics like Hamiltonian dynamics?
I would write:
Given a Hamiltonian flow map $\Phi^t:\Phi^t(x(0)) \rightarrow x(t)$, with $x=\lbrace q,p \rbrace$, the closed orbit with period $T$ defined by $O_x=\lbrace x, S^{\Delta t}(x), ...,S^T(x) \rbrace$ ,
where $\Delta t$ is a small timestep, supports the measure
$\delta_{O_x} = \frac{1}{T} \int_{0}^T \delta(\Phi^t(x)) \mathrm{d}t$ .
Is this correct?
References [1] http://wwwf.imperial.ac.uk/~mrasmuss/ergodictheory/ErgodicTheoryNotes.pdf pag.38