Multivariable Complex Dynamics

Broadly speaking, the object in ‘multivariable complex dynamics’ is to understand dynamical properties of holomorphic (and more generally meromorphic) mappings f : X → X on compact complex manifolds X with complex dimension two or larger. The subject has been actively pursued for about twenty years now, and it has its genesis in several sources. First among these is the remarkable success in understanding dynamics of (single variable) holomorphic maps f : P1 → P1 on the Riemann sphere, and in particular of polynomial maps f : C2 → C2. In the 1980’s it was noted [42] that the Hénon mappings (x, y) "→ (y, y2 + c − ax) are natural two dimensional generalizations of quadratic polynomials z "→ z2+c of a single variable. More generally, people began to ask whether and how certain natural and well-understood concepts for holomorphic maps of one variable might generalize to higher dimensions. For example, people wondered what the appropriate definition of Fatou and Julia (i.e. stable and unstable) set might be for a holomorphic map f : Pn → Pn on higher dimensional projective space [37]; similarly, there were questions about the local behavior of a multi-variable map near a superattracting or indifferent fixed point. Somewhat apart from such considerations, there was a remarkable and simple argument of Gromov [41] that gave the precise value for the topological entropy of a holomorphic map f : Pn → Pn in terms of the degrees of the homogeneous polynomials defining f . The variational principle from smooth dynamics states that topological entropy is the supremum of the various metric entropies with respect to f -invariant measures. Hence it was logical to investigate questions concerning existence, uniqueness and structure of f -invariant measures that maximize the metric entropy of a holomorphic map f . Finally, it was realized that recent developments in several complex variables and dynamical systems, particularly those surrounding pluripotential theory and the theory of non-uniformly hyperbolic systems, might furnish sufficient technical tools for addressing dynamical issues like the ones described above. Two decades away from its beginning, the field of multivariable complex dynamics continues to flourish and grow steadily. Some of the initial questions (e.g. existence and uniqueness of measures of maximal entropy) are much better understood now, having been answered for broad classes of holomorphic mappings. Others have seen less definitive progress, and of course, the list of questions that motivated the subject initially has greatly expanded. So too has the range of techniques that are employed to address them. In particular, complex geometry now plays a large role. Some of the more recently employed tools (e.g. laminar currents, superpotentials, and valuative analysis) did not exist or were not nearly as refined when the subject began. Rather, they were developed in direct response to dynamical questions, and now seem likely to find application in the larger world of mathematics outside complex dynamics. Our goal in the remainder of this exposition is to describe some of these developments, particularly those related to our March 2009 conference at Banff International Research Station, in more detail below. Though