Hénon maps: a list of open problems
We propose a set of questions on the dynamics of Hénon maps from the real, complex, algebraic and arithmetic points of view.
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arXiv subjects
Eric Bedford
Publications and source records attributed to Eric Bedford.
Topological and geometric hyperbolicity criteria for polynomial automorphisms of C^2
We prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of C^2. Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.
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Hyperbolicity and Quasi-hyperbolicity in Polynomial Diffeomorphisms of ${\Bbb C}^2$
We consider complex Henon maps which are quasi-hyperbolic. We show that a quasi-hyperbolic map is uniformly hyperbolic if and only if there are no tangencies between stable and unstable manifolds.
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Julia sets for polynomial diffeomorphisms of ${\Bbb C}^2$ are not semianalytic
For any polynomial diffeomorphism $f$ of ${\Bbb C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is semi-analytic.
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No smooth Julia sets for polynomial diffeomorphisms of $\mathbb{C}^2$ with positive entropy
For any polynomial diffeomorphism $f$ of $\mathbb{C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is $C^1$ smooth as a manifold-with-boundary.
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Geometric proof of the $λ$-lemma
We give a geometric approach to the proof of the $λ$-lemma. In particular, we point out the role pseudoconvexity plays in the proof.
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Dynamics of Polynomial Diffeomorphisms of $C^2$: Foliations and laminations
This essay summarizes the state of the art on some aspects of the dynamics of polynomial diffeomorphsms in complex dimension two, and it presents a number of open questions.
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Invertible dynamics on blow-ups of P^k
We discuss automorphisms and pseudo-automorphisms on blowups of complex projective space with an eye to finding ones with interesting dynamical behavior.
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A Symbolic Characterization of the Horseshoe Locus in the Hénon Family
We consider the family of quadratic Hénon diffeomorphisms of the plane ${\bf R}^2$. A map will be said to be a "horseshoe" if its restriction to the nonwandering set is hyperbolic and conjugate to the full 2-shift. We give a criterion for being a horseshoe based on an auxiliary coding which describes positions of points relative to the stable manifold of one of the fixed points. In addition we describe the topological conjugacy type of maps on the boundary of the horseshoe locus. We use complex techniques and we work with maps in a parameter region which is a 2-D analog of the familiar "${1\over 2}$-wake" for the quadratic family $p_c(z) = z^2$.
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Pseudoautomorphisms with invariant elliptic curves
We describe an explicit method for constructing pseudo-automorphisms of a space $X$ which is obtained by blowing up points of $P^k$ (or a product $P^k \times \cdots \times P^k$). The centers of blowup are chosen to lie on an elliptic normal curve and are determined using the arithmetic on the curve. These pseudo-automorphism have dynamical degree greater than $1$.
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Pseudo-automorphisms with no invariant foliation
We construct an example of a birational transformation of a rational threefold for which the first and second dynamical degrees coincide and are $>1$, but which does not preserve any holomorphic (singular) foliation. In particular, this provides a negative answer to a question of Guedj. On our way, we develop several techniques to study foliations which are invariant under birational transformations.
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Parabolic Bifurcations in Complex Dimension 2
Parabolic bifurcations in one complex dimension demonstrate a wide variety of interesting dynamical phenomena. In this paper we consider parabolic bifurcations of families of diffeomorphisms in two complex dimensions. Specifically we consider a two variable family of diffeomorphisms $F_ε: M\to M$ given locally by $$F_ε(x,y) = (x + x^2 + ε^2+ ..., b_εy+...)$$ where $|b_ε|<1$, and the `$...$' terms involve $x$, $y$ and $ε$.
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Dynamics of (Pseudo) Automorphisms of 3-space: Periodicity versus positive entropy
We study the iterative behavior of the family of 3-step linear fractional recurrences and the family of birational maps they define. We determine all the possible periodicities within this family or, equivalently, the birational maps of finite order. This family also contains pseudo-automorphisms of infinite order. One such family consists of completely integrable maps, and another family consists of maps of positive entropy. Both of these families have invariant families of $K3$ surfaces.
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The Dynamical Degrees of a Mapping
Let f be a rational mapping of a space X . The complexity of (f,X) as a dynamical system is measured by the dynamical degrees $δ_p(f)$, $1\le p\le {\rm dim}(X)$. We give the definition of the dynamical degrees show how they are computed in certain cases. For instance, we show that if the dynamical degree of an automorphism of a Kähler manifold is greater than one, then it must be irrational.
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Linear Fractional Recurrences: Periodicities and Integrability
We consider k-step recurrences of the form $z_{n+k} = A(z)/B(z)$, where A and B are linear functions of $z_n, z_{n+1}, ..., z_{n+k-1}$, which we call k-step linear fractional recurrences. The first Theorem in this paper shows that for each k there are k-step linear fractional recurrences which are periodic of period 4k. Among this class of recurrences, there is also the so-called Lyness process, which has the form $A(z)/B(z) = (a +z_{n+1} + z_{n+2} + ... + z_{n+k-1})/z_n$. The second Theorem shows that the Lyness process has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.
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Dynamics of Rational Surface Automorphisms: Rotation Domains
We consider rational surface automorphisms with positive entropy. A Fatou component is said to be a rotation domain if the automorphism induces a torus action on it. Here we construct a rational surface automorphism with positive entropy with the following property: it has a rotation domain which contains both a curve of fixed points and isolated fixed points. This Fatou component cannot be imbedded into complex euclidean space, so we introduce a global linear model space and show that it can be globally linearized in this model.
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Degree Complexity of Matrix Inversion
For a q by q matrix x=(x_{i,j}) we let J(x)=(x_{i,j}^{-1}) be the Hadamard inverse, which takes the reciprocal of the elements of x . We let I(x)=(x_{i,j})^{-1} denote the matrix inverse, and we define K=I\circ J to be the birational map obtained from the composition of these two involutions. We consider the iterates K^n=K\circ...\circ K and determine degree complexity of K, which is the exponential rate of degree growth of the degrees of the iterates.
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Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences
We consider the family $f_{a,b}(x,y)=(y,(y+a)/(x+b))$ of birational maps of the plane and the parameter values $(a,b)$ for which $f_{a,b}$ gives an automorphism of a rational surface. In particular, we find values for which $f_{a,b}$ is an automorphism of positive entropy but no invariant curve. The Main Theorem: If $f_{a,b}$ is an automorphism with an invariant curve and positive entropy, then either (1) $(a,b)$ is real, and the restriction of $f$ to the real points has maximal entropy, or (2) $f_{a,b}$ has a rotation (Siegel) domain.
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