Editing Complex dynamics (section)

===Kummer automorphisms===
Some abelian varieties have an automorphism of positive entropy. For example, let ''E'' be a complex [[elliptic curve]] and let ''X'' be the abelian surface <math>E\times E</math>. Then the group <math>GL(2,\mathbf{Z})</math> of invertible <math>2\times 2</math> integer matrices acts on ''X''. Any group element ''f'' whose [[trace (linear algebra)|trace]] has absolute value greater than 2, for example <math>\begin{pmatrix}2&1\\1&1\end{pmatrix}</math>, has spectral radius greater than 1, and so it gives a positive-entropy automorphism of ''X''. The equilibrium measure of ''f'' is the Haar measure (the standard Lebesgue measure) on ''X''.<ref>Cantat & Dupont (2020), section 1.2.1.</ref>

The '''Kummer automorphisms''' are defined by taking the quotient space by a finite group of an abelian surface with automorphism, and then [[blowing up]] to make the surface smooth. The resulting surfaces include some special K3 surfaces and rational surfaces. For the Kummer automorphisms, the equilibrium measure has support equal to ''X'' and is [[smooth function|smooth]] outside finitely many curves. Conversely, Cantat and Dupont showed that for all surface automorphisms of positive entropy except the Kummer examples, the equilibrium measure is not absolutely continuous with respect to Lebesgue measure.<ref>Cantat & Dupont (2020), Main Theorem.</ref> In this sense, it is usual for the equilibrium measure of an automorphism to be somewhat irregular.