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Jeffrey Diller

Publications and source records attributed to Jeffrey Diller.

At least 19 recordsLinked to original sources
arXiv2026

Nonexistence of degree two rational multisections of conic bundles over the plane

We prove that a standard conic bundle $X \to \mathbb P^2_{\mathbb C}$ whose discriminant is very general of degree at least 18 admits no rational multisections of degree two. This is the first step towards proving a conjecture of Iskovskikh that there are conic bundle threefolds that are not unirational, since to prove that $X$ is not unirational, it suffices to show that there are no rational multisections of any degree. Proving Iskovskikh's conjecture would provide the first example of a rationally connected variety that is not unirational.

math.AG↗
arXiv2025

Energy, equilibrium measure and entropy for toric surface maps

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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arXiv2024

Equidistribution without stability for toric surface maps

We prove an equidistribution result for iterated preimages of curves by a large class of rational maps $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP}^2$ that cannot be birationally conjugated to algebraically stable maps. The maps, which include recent examples with transcendental first dynamical degree, are distinguished by the fact that they have constant Jacobian determinant relative to the natural holomorphic two form on the algebraic torus. Under the additional hypothesis that $f$ has "small topological degree'' we also prove an equidistribution result for iterated forward images of curves. To prove our results we systematically develop the idea of a positive closed $(1,1)$ current and its cohomology class on the inverse limit of all toric surfaces. This, in turn, relies upon a careful study of positive closed $(1,1)$ currents on individual toric surfaces. This framework may be useful in other contexts.

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arXiv2023

Birational maps with transcendental dynamical degree

We give examples of birational selfmaps of $\mathbb{P}^d, d \geq 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation.

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arXiv2018

Entropy of real rational surface automorphisms

We compare real and complex dynamics for automorphisms of rational surfaces that are obtained by lifting \chg{some} quadratic birational maps of the plane. In particular, we show how to exploit the existence of an invariant cubic curve to understand how the real part of an automorphism acts on homology. We apply this understanding to give examples where the entropy of the full (complex) automorphism is the same as its real restriction. Conversely and by different methods, we exhibit different examples where the entropy is strictly decreased by restricting to the real part of the surface. Finally, we give an example of a rational surface automorphism with positive entropy whose periodic cycles are all real.

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arXiv2016

Typical dynamics of plane rational maps with equal degrees

Let $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP^2}$ be a rational map with algebraic and topological degrees both equal to $d\geq 2$. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms $T:\mathbb{CP}^2\to\mathbb{CP}^2$, the perturbed map $T\circ f$ admits exactly two ergodic measures of maximal entropy $\log d$, one of saddle and one of repelling type. Neither measure is supported in an algebraic curve, and $T\circ f$ is `fully two dimensional' in the sense that it does not preserve any singular holomorphic foliation. Absence of an invariant foliation extends to all $T$ outside a countable union of algebraic subsets. Finally, we illustrate all of our results in a more concrete particular instance connected with a two dimensional version of the well-known quadratic Chebyshev map.

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arXiv2014

Rational surface maps with invariant meromorphic two forms

We consider a rational map f:S->S of a complex projective surface together with an invariant meromorphic two form. Under a mild topological assumption on the map, we show that the zeroes of the invariant form can be eliminated by birational change of coordinate. In this context, when the form has no zeroes, we investigate the notion of algebraic stability for f. We show in particular that algebraic stability is equivalent to a more tractable condition involving the behavior of f on the poles of the form. Finally, we illustrate our results in the particular case where S is the projective plane and the invariant form is dx dy / xy, showing that our criterion for stability translates to whether or not the rotation number for a certain circle homeomorphism is rational.

math.AG↗
arXiv2010

Cremona transformations, surface automorphisms and plane cubics

We give a method for constructing many examples of automorphisms with positive entropy on rational complex surfaces. The general idea is to begin with a quadratic Cremona transformation that fixes a reduced cubic curve and then use the group structure on the cubic to understand when the indeterminacy and exceptional behavior of the transformation may be eliminated by repeated blowing up.

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arXiv2009

Dynamics of meromorphic maps with small topological degree II: Energy and invariant measure

We continue our study of the dynamics of meromorphic mappings with small topological degree on a compact Kähler surface $X$. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and admits a natural geometric description. Our hypotheses are always satisfied when $X$ has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of $\mathbf{C}^2$. They are new even in the birational case. We also exhibit families of mappings where our assumptions are generically satisfied and show that if counterexamples exist, the corresponding measure must give mass to a pluripolar set.

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arXiv2009

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

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.

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arXiv2009

Dynamics of meromorphic maps with small topological degree I: from cohomology to currents

We consider the dynamics of a meromorphic map on a compact kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which its iterates expand the cohomology class of a kahler form. Our goal in this article and its sequels is to carry out a conjectural program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, converting information about the linear action of the map on cohomology to invariant currents with special geometric structure. We also give some examples and identify some additional properties of maps on irrational surfaces and of maps whose invariant cohomology classes have vanishing self-intersection.

math.CV↗
arXiv2008

Regularity of Dynamical Green Functions

For meromorphic maps of complex manifolds, ergodic theory and pluripotential theory are closely related. In nice enough situations, dynamically defined Green's functions give rise to invariant currents which intersect to yield measures of maximal entropy. `Nice enough' is often a condition on the regularity of the Green's function. In this paper we look at a variety of regularity properties that have been considered for dynamical Green's functions. We simplify and extend some known results and prove several others which are entirely new. We also give some examples indicating the limits of what one can hope to achieve in complex dynamics by relying solely on the regularity of a dynamical Green's function.

math.CV↗
arXiv2005

Dynamics of a two parameter family of plane birational maps: maximal entropy

We consider the real dynamics of a two parameter family of plane birational maps, focusing especially on an open subset of parameter space on which the real and complex dynamics are in close agreement. On the complex side, we find a rational complex surface to which the maps extend in a well-defined fashion and then calculate the induced action on the Picard group of the surface. On the real side, we use the critical sets of the maps to produce a combinatorial model for the dynamics. By comparing the two points of view, we are able to code points in the real non-wandering set, describe the behavior of wandering orbits, and identify a measure of maximal entropy for the real dynamics.

math.DS↗
arXiv2005

Invariant curves for birational surface maps

We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and number of irreducible components of the curve. In the case of an invariant curve with genus equal to one, we show that there is an associated invariant meromorphic two form.

math.AG↗
arXiv2003

Energy and Invariant Measures for Birational Surface Maps

When a birational surface map is expanding on cohomology there is a canonical way to associate positive closed currents to the map and its inverse. In this paper we use a version of Dirichlet energy to construct the wedge product of these two currents under a very weak additional condition on the map. We show that the resulting measure is invariant and mixing, and we establish that its Lyapunov exponents are finite and non-zero.

math.CV↗