arXiv subjects
Publications and source records attributed to Eric Bedford.
We construct k-parameter families of rational surface automorphisms for any k. These are automorphisms of surfaces X, which are constructed from iterated blowups over the projective plane. In certain cases: we are able to determine the exact automorphism group of X, as well as when two of the surfaces X are inequivalent.
We compute the degree complexity of a family of birational mappings of the plane with high order singularities.
Let $A$ be an integer matrix, and let $f_A$ be the associated monomial map. We give a connection between the eigenvalues of $A$ and existence of a linear recurrence relation in the sequence of degrees.
We show how a variant of the Lefschetz Fixed Point Theorem may be used to count the number of periodic orbits for certain rational difference equations.
Continuing work begun in a previous paper, we study the real dynamics of a family of plane birational maps. This time we consider a parameter range where the real and complex dynamics are different. We show in particular that for one parameter value, the real dynamics are transient even though the map is not integrable.
We give a formula for the complex Monge-Ampere operator applied to the maximum of a finite number of functions.
We consider two (densely defined) involutions on the space of $q\times q$ matrices; $I(x_{ij})$ is the matrix inverse of $(x_{ij})$, and $J(x_{ij})$ is the matrix whose $ij$th entry is the reciprocal $x_{ij}^{-1}$. Let $K=I\circ J$. The set ${\cal SC}_q$ of symmetric, cyclic matrices is invariant under $K$. In this paper, we determine the degrees of the iterates $K^n=K\circ...\circ K$ restricted to ${\cal SC}_q$.
We consider the set of all 2-step recurrences (difference equations) that are given by linear fractional maps. These give birational maps of the plane. We determine the degree growth of these birational maps. We find the all the maps in this family that are periodic. This also leads to new surface automorphisms with positive 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.
We consider the complexification of the Henon family of quadratic diffeomorphisms of the plane, and the region of parameter space corresponding to complex horseshoes. We discuss some conjectures about the global topology of the complex horseshoe locus.
Let $f$ be a birational map of ${\bf C}^d$, and consider the degree complexity, or asymptotic degree growth rate $δ(f)=\lim_{n\to\infty}({\rm deg}(f^n))^{1/n}$. We introduce a family of elementary maps, which have the form $f=L\circ J$, where $L$ is (invertible) linear, and $J(x_1,...,x_d)=(x_1^{-1},...,x_d^{-1})$. We develop a method of regularization and show how it can be used to compute $δ$ for an elementary map.
This paper concerns the Henon family of diffeomorphisms of the plane for which the absolute value of the jacobian parameter is bounded by .08. We describe the parameter locus for which the mapping is a horseshoe.
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.
We give a detailed description of the dynamics of a family of birational mappings of the plane. The organizing theme of our analysis is that this family is essentially conjugate to the golden mean subshift.
This paper continues our investigation of the dynamics of polynomial diffeomorphisms of C^2. We introduce a dynamical property of polynomial diffeomorphisms that generalizes hyperbolicity in the way that semi-hyperbolicity generalizes hyperbolicity for polynomial maps of the complex plane.
In this paper we use complex techniques to study the structure of real Henon diffeomorphisms of maximal topological entropy.
We study the dynamics of polynomial mappings f:C^k to C^k of degree at least 2 that extend continuously to projective space P^k. Our approach is to study the dynamics near the hyperplane at infinity and then making a descent to K --- the set of points with bounded orbits --- via external rays.
Given a polynomial diffeomorphism f: C^2 -> C^2 there is a set $J_f\subset{\bf C}^2$ which we call the Julia set of f. The set $J_f\subset C^2$ plays the role of the Julia set $J\subset{\bf C}$ for a polynomial map of C. In the study of polynomial maps of C a great deal of attention has been paid to the connectivity of the Julia set. The focus of this paper is to investigate the J-connected/J-disconnected dichotomy in the case of polynomial diffeomorphisms of C^2. The Jacobian determinant of f is constant. We make the standing assumption that $|det\ Df|\le 1$ (this can always be achieved by replacing f by $f^{-1}$ if necessary). The set $J^-$ is the set of points with bounded backward orbits. The set $U^+$ is the set of points with unbounded forward orbits. Let p be a periodic saddle point and let $W^u(p)$ be its unstable manifold. The set $W^u(p)$ will be a Riemann surface conformally equivalent to C. Theorem 1. The following are equivalent: 1. For some periodic saddle point p, some component of $W^u(p)\cap U^+$ is simply connected. 2. The set $J^-\cap U^+$ has a lamination by simply connected leaves so that for any periodic saddle point p each component of $W^u(p)\cap U^+$ is a leaf of this lamination. 3. For any periodic saddle point p, each component of $W^u(p)\cap U^+$ is simply connected. If f satisfies one of these conditions we say that f is unstably connected. Theorem 2. The set J is connected if and only if f is unstably connected. These results imply that we can determine the connectivity of J by considering the forward orbits of points in a single unstable manifold. These results open the door to computer exploration of the topology of two dimensional Julia sets and the connectivity locus in the parameter space.