I'm interesting in find (if it exists) any manifold with volume form and any diffeomorphism on it such that for any ball B corresponding sequence of Biterated volumes (just measures of $T^i(B)$, where T is diffeomorphism) may take values arbitrarily close to volume of all manifold. In compact manifold case this question are equivalent to arbitary close deflation of ball in such sequence. What for noncompact manifolds in this case? What for oneparametric group of diffeomorphisms? More generalized, how invariant measures far from initial category/conditions? What is degree of measure invariance or measurepreservation of our diffeomorphism (except divergence if it come from vector field; some global properties, please)? Sorry for bad english.

As far as I understand your question, examples you want are provided by the socalled NorthSouth dynamical systems, which are actions of $\mathbb Z$ (discrete time) or $\mathbb R$ (continuous time) with one repelling and one attractive fixed points such that everything else is contracted to the attractive point as time goes to $+\infty$ and to the repelling point as time goes to $\infty$. For instance, any hyperbolic isometry of the hyperbolic plane generates this kind of action on the boundary circle. 


Not sure if it is related, but in Lemma 1 of this paper (A.Avila, J. Bochi, A C1 generic map has no invariant absolutely continuous probability measure) it is proved that if a map has no invariant absolutely continuous probability measure, then, there exists a compact set $K$ of measure abitrarily close to $1$ which has an iterate with arbitrarily small measure. Sorry if this has nothing to do, but from how I understood the question, at least this should be useful. EDIT: Rereading the question, I've noticed that you are maybe more concerned about iterates of balls. For this, it depends on what you want. For example, if you want, for any $\varepsilon$ a ball of smaller radius that has iterates with volume close to the manifold, it is easy to get (say, for example a northsouth map). I believe this should be not what you are looking for. If you look for every ball at a time, this won't be true in general, and let me rephrase your question (or at least the one I am responding to): Is there a homeomorphism $f$ such that given a ball $B$ and $\varepsilon>0$ there exists an integer $n$ such that $f^n(B)$ has measure larger than $1\varepsilon$. If $f$ is not transitive (or at least, if it has at least two chain recurrence classes), then, there exists an open set such that $\overline{U} \subset f(U)$ so, if you consider a ball in $\overline{U}^c$ it will never reach measure bigger than the complement of $U$. For $f$ transitive, there are also examples where this does not hold. For example, for conservative $f$, it trivially does not hold. In other cases, I would presume that it is not true (at least for ``tipical'' diffeomorphisms), however, I don't know a proof nor an example. 


I would try to reanimate this thread and write pair of trivial views. first, on 1dimensional manifold it's imposible just because any suitable diffeomorphisms comes from vector fields (such topic is rised on mathoverflow) and last have potential on universe cover  line, but if potential exist problem have no solution (on any dimensional manifold). On other hand it's not surprisingly and now my main goal is to illustrate that in other cases existence of such diffeo intuitively justified. Call diffeo of M to be stransitive if his natural action on any selfproducts M ($M\times M\ldots\times M$) is topological transitive. In other words any number of pairdistinct points on M is to posible send arbitrarily close to any else set of such number points. So images by iterations of any fixed ball is arbitrarily dense in manifold. Intuitively speaking, obstruction to volume changes high lies in ability us diffeo change curvature of boundary strongly enough. I hope to get around this. Any ideas? 


Maybe try to solve a little bit simpler problem: is there a manifold and a diffeomorphism such that any small enough ball will have its volume increased at least 100 times (or any constant big number) under diffeomorhism, iterated some times. 

