The paper proves equidistribution of preimages of curves for generic finitely supported random walks of the Cremona group on the inverse limit of spaces of currents across models of the plane.
Energy, equilibrium measure and entropy for toric surface maps
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abstract
We consider the ergodic theory of plane rational maps that preserve the natural holomorphic volume form on the algebraic torus. Specifically we construct natural invariant probability measures for a large class of such maps by intersecting the equilibrium currents we constructed in our previous work [DR]. We show further that these measures are mixing and that each admits an underlying geometric product structure. The main result of [DDG3] then implies that the topological entropy of each map covered by our results is the log of its first dynamical degree. In light of examples presented in [BDJ], this implies in particular that the entropy of a rational map can equal the log of a transcendental number.
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Random products of birational maps: Equidistribution of preimages of curves
The paper proves equidistribution of preimages of curves for generic finitely supported random walks of the Cremona group on the inverse limit of spaces of currents across models of the plane.
- Complex dynamics perspective for birational maps of the plane arising from cluster algebra mutations