  We introduce an entropy analysis of time series, repeated measurements of
statistical observables, based on an Eulerian homogeneous degree-one entropy
function $\Phi(t,n)$ of time $t$ and number of events $n$. The duality of
$\Phi$, in terms of conjugate variables $\eta=-\Phi'_t$ and $\mu=\Phi'_n$,
yields an ``equation of state'' (EoS) in differential form that resembles the
Gibbs-Duhem relation in classical thermodynamics: $t d\eta-n d\mu = 0$. For
simple Poisson counting with rate $r$, $\eta=r(e^{\mu}-1)$. The conjugate
variable $\eta$ is then identified as being equal to the Hamiltonian function
in a Hamilton-Jacobi equation for $\Phi(t,n)$. Applying the same logic to the
entropy function of time correlated events yields a Hamiltonian as the
principal eigenvalue of a matrix. For time reversible case it is the sum of a
symmetric Markovian part $\sqrt{\pi_i}q_{ij}/\sqrt{\pi_j}$ and the conjugate
variables $\mu_i\delta_{ij}$. The eigenvector, as a posterior to the naive
counting measure used as the prior, suggests a set of intrinsic characteristics
of Markov states.
