Jeffrey Diller

Energy and invariant measures for birational surface maps

Given a birational self-map of a compact complex surface, it is useful to find an invariant measure that relates the dynamics of the map to its action on cohomology. Under a very weak hypothesis on the map, we show how to construct such a measure. The main point in the construction is to make sense of the wedge product of two positive, closed (1, 1)-currents. We are able to do this in our case because local potentials for each current have “finite energy” with respect to the other. Our methods also suffice to show that the resulting measure is mixing, does not charge curves, and has nonzero Lyapunov exponents.

Invariant curves for birational surface maps

We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and number of irreducible components of the curve. In the case of an invariant curve with genus equal to one, we show that there is an associated invariant meromorphic two-form.

Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure

We continue our study of the dynamics of meromorphic mappings with small topological degree ?2(f)<?1(f) on a compact Kahler surface X. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and has a natural geometric description. Our hypotheses are always satisfied when X has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of C2. They are new even in the birational case (?2(f)=1). We also exhibit families of mappings where our assumptions are generically satisfied and show that if counterexamples exist, the corresponding measure must give mass to a pluripolar set.

Real and complex dynamics of a family of birational maps of the plane: The golden mean subshift

We describe the (real) dynamics of a family of birational mappings of the plane. By combining complex intersection theory and techniques from smooth dynamical systems, we are able to give an essentially complete account of the behavior of both wandering and nonwandering orbits. In particular, the golden mean subshift provides a topological model for the dynamics on the nonwandering set. While the mappings are not hyperbolic, they are shown to possess many of the structures associated with hyperbolicity.

Topological entropy on saddle sets in P2

We consider hyperbolic sets of saddle type for holomorphic mappings in P. Our main result relates topological entropy on such sets to a normal families condition on local unstable manifolds.

Calculation of Slow Invariant Manifolds for Reactive Systems

One-dimensional slow invariant manifolds for dynamical systems arising from modeling unsteady, isothermal, isochoric, spatially homogeneous, closed reactive systems are calculated. The technique is based on global analysis of the composition space of the reactive system. The identification of all the system’s finite and infinite critical points plays a major role in calculating the system’s slow invariant manifold. The slow invariant manifolds are constructed by calculating heteroclinic orbits which connect appropriate critical points to the critical point which corresponds to the unique stable physical critical point of chemical equilibrium. The technique is applied to small and large detailed kinetics mechanisms for hydrogen combustion.