Pullback of currents by meromorphic maps

Abstract

My dissertation researches on properties of iterations $f^n=f\circ f\circ \ldots \circ f$ ($n$ times) of a selfmap $f:X\rightarrow

X$. Here $X$ is a compact K\"ahler manifold and $f$ is a dominant meromorphic map. For holomorphic maps, a variational principle for smooth maps proves the existence of a measure which is invariant under $f$ and has maximal entropy (i.e. the entropy of the measure equals the topological entropy). The same question is harder to answer for a general meromorphic map $f$, due to the fact that $f$ is not continuous. Since the pioneer work of Bedford et al. on H\'enon maps, a common strategy is to first establish the existence of appropriate invariant currents (a generalization of

measures), and then use them to construct invariant measures. To this end, it is important to know what currents can be pulled back or pushed forward by a map $f$. In my dissertation, based upon a regularization theorem of Dinh and Sibony, I give a definition of pulling back by a given meromorphic map for a large class of currents. This pullback operator is compatible with the definitions given by many other authors. Many applications and examples are given.