Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory

By Jeffrey Diller, Romain Dujardin, Vincent Guedj. 2008

47 citations recorded by the source corpus.

We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the invertible case and is, to our knowledge, one among not very many instances in which a natural invariant measure for a non-invertible dynamical system is well-understood.

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/434184e66f1af3e9a1722d50978dbeacfd46267e

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In Condensed Matter Physics

Xie Junyi

In Mathematical Physics

Citations in the corpus

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].”
Equidistribution problems of complex dynamics in higher dimension (2016, background): “The reader will find more results for surfaces in [16, 39, 58] which are too technical to state in this paper.”
Periodic points of birational transformations on projective surfaces (2015, methodology): “ : X 99K X be a birational map. If λ1(f) > 1 then the set of non-critical periodic points is Zariski dense. In the case k = C, this theorem has been proved in many cases using analytic methods. In [11] of Diller, Dujardin and Guedj, they proved it for birational polynomial maps; or more generally”