Energy and invariant measures for birational surface maps
By Eric Bedford, Jeffrey Diller. 2003
74 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/e852b54766f6c845934756dd254e9afe1e8bcf38
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In Geometry and Topology
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Dynamical degrees of birational transformations of projective surfaces (2013, background): “If k is the field of complex numbers, the neperian logarithm log(λ(f)) provides an upper bound for the topological entropy of f : X(C) X(C) and is equal to it under natural assumptions (see [3,23]).”
- Periodic points of birational transformations on projective surfaces (2015, background): “d it for birational polynomial maps; or more generally for any birational map such that the points of indeterminacy of f−1 does not cluster too much near the points of indeterminacy of f. We refer to [3] for a precise statement. It follows from [12] that any birational map with λ1 > 1 defined over”
- Dynamics of automorphisms of compact complex surfaces (2014, background): “Then pluri-potential theory can be applied successfully to construct good f -invariant laminar currents and the unique measure of maximal entropy; this measure describes the distribution of periodic points: See [6] and [66], as well as [50, 51, 52].”