I just know very little about currents but I need vexedly. Thanks for your help. 


Let $M$ be a closed orientable surface and $I=(f\_t)\_{t\in[0,1]}$ be an isotopies from identity to $f$. Suppose that $\mu$ is a probability measure on $M$ and $f$ preserves the measure $\mu$. Let $I.z$ be the oriented trajectory of $z$. We can define the 1-current [I.z] (if $I$ is smooth and $I.z$ is simple, it is a 1-sub-manifold of $M$).
If $\rho\_I(\mu)=\int\_M [I.z] \, d\mu$=0, we call $I$ is $\mu$-hamiltonian. 
My question is that the space of 1-currents is finite dimension? And the first current homology is isomorphic to the standard singular homology $H\_1(M,\mathbb{R})$? Further, can we use the Birkhoff Ergodic theory to the space of 1-currents? I mean, for example, is the following limit
$$\frac{1}{n}\sum\_{i=0}^{n-1}[I.(f^i(z))]$$ $\mu$-a.e. exists as $n\rightarrow+\infty$? 



The currents homology of closed orientable surfaces and Birkhoff Ergodic theorem?

> 
> My question is that the space of 1-currents is finite 
>  dimension? 
> 
> 
> 


There are several versions of currents. The most standard definition is
the dual space of the space of k-forms of certain regularity. It is
definitely infinite-dimensional, except in some very degenerate cases (e.g. dimension of the space is 0)


Roughly-speaking 1-current is something
you can integrate against 1-forms, e.g. any formal linear combination of
oriented 1D submanifolds will do or, more generally, 1D submanifolds with a
measures on them, tangent vector-fields with measure-coefficients, etc.



> 
> And the first current homology is isomorphic to the 
>  standard singular
>  homology $H\_1(M,\mathbb{R})$?
> 
> 
> 


Yes, once you use definitions, that turn spaces of currents into a complex.


If trajectory $I.z$ is not rectifiable, then, strictly speaking, it does not
give you a 1-current. On the other hand, it seems, that you only need
the homology class of the rotation vector $\rho\_I(\mu)$ to be 0, not $\rho$ itself. Is it right?
If so, would the modified definition work for your purposes?
$$
 \rho\_I(\mu) := \int\_{M} [\gamma\_{z}] d\mu
$$
where $\gamma\_{z}$ is the **unique** geodesic from $z$ to $f(z)$ homotopic to
$I.z$ rel endpoints, taken with
respect to some fixed auxiliary metric of non-positive curvature on $M$. If $I.z$ were rectifiable, then $([I.z]-[\gamma\_{z}])$ is
a boundary, thus new definition of $\rho$ preserves its homology class in that case. You have to check, that $z\mapsto[\gamma\_z]$ is measurable, which, I believe it is, because the space of geodesic intervals is in one-to-one correspondence with pairs of points, (up to deck transformations) in the universal cover of $M$ in a measurable way.


For the same reason, I believe, (but did not verify) that the Birkhoff theorem will hold once the summands are changed to $[\gamma\_{f^i(z)}]$ and the considered space of currents is complete.


