Abstract
Dynamical degrees and spectra can serve to distinguish birational automorphism groups of varieties in quantitative, as opposed to only qualitative, ways. We introduce and discuss some properties of those degrees and the Cremona degrees, which facilitate computing or deriving inequalities for them in concrete cases: (generalized) lower semi-continuity, sub-multiplicativity, and an analogue of Picard-Manin/Zariski-Riemann spaces for higher codimension cycles. We also specialize to cubic fourfolds and show that under certain genericity assumptions the first and second dynamical degrees of a composition of reflections in points on the cubic coincide.
Highlights
1 Introduction For a smooth projective variety X of dimension n and a birational self-map f : X X, one can define a tuple of real numbers λ0(f ) = 1, λ2(f ), . . . , λn−1(f ), λn(f ) = 1, where λi(f ) ≥ 1 for 2 ≤ i ≤ n − 1, called the dynamical degrees of f
The dynamical degrees turn out to be invariant under birational conjugacy, so that the dynamical spectrum
Are some ideas how the spectra might differ: (1) As point sets, that is, there might be a tuple λ(f ) in the spectrum of P4 which is not in the spectrum of X. This could, for example, be proven if one could show that on X the dynamical degrees have to satisfy other additional inequalities, coming from the geometry of X, which can be violated on P4
Summary
The number fields generated by each tuple of dynamical degrees on X might differ from the ones for P4. This is a sample of a type of result which says that special dynamical degrees can only arise in the presence of special dynamics. The graph Γf ⊂ X × Y of f is the closure of the locus of points (x, f (x)) with x ∈ dom(f )
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