Parabolic bifurcations in one complex dimension demonstrate a wide variety of interesting dynamic... more 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 $F_\epsilon: M\to M$ given locally by $$F_\epsilon(x,y) = (x + x^2 + \epsilon^2+ ..., b_\epsilon y+...)$$ where $|b_\epsilon|<1$, and the `$...$' terms involve $x$, $y$ and $\epsilon$.
We consider the family of quadratic H\'enon diffeomorphisms of the plane ${\bf R}^2$. A map w... more We consider the family of quadratic H\'enon 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 $p_c(z) = z^2$.
For any polynomial diffeomorphism $f$ of $\mathbb{C}^2$ with positive entropy, neither the Julia ... more 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.
We consider the real dynamics of a two parameter family of plane birational maps, focusing especi... more 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
Page 1. BOUNDARY CONTINUITY OF PROPER HOLOMORPHIC CORRESPONDENCES Eric Bedford Indiana University... more Page 1. BOUNDARY CONTINUITY OF PROPER HOLOMORPHIC CORRESPONDENCES Eric Bedford Indiana University Bloomington, Indiana 47405Steve Bell Princeton University Princeton, New Jersey 08544 i. Introduction ...
... for some neighborhood W of Page 4. 262 ERIC BEDFORD and JOHN ERIK FORN^SS z, Vz П W may be ob... more ... for some neighborhood W of Page 4. 262 ERIC BEDFORD and JOHN ERIK FORN^SS z, Vz П W may be obtained as the union of all real analytic curves in W starting at z whose tangents lie in H (oil). Since d il is real analytic ...
It is proved that proper holomorphic mappings between n-dimensional bounded pseudoconvex domains ... more It is proved that proper holomorphic mappings between n-dimensional bounded pseudoconvex domains with real analytic boundaries extend holomorphically past the boundary whenever the target domain is strictly pseudoconvex.
ABSTRACT It is shown that if a bounded domain \Omega\subset\mathbf{C}^2 with real analytic bounda... more ABSTRACT It is shown that if a bounded domain \Omega\subset\mathbf{C}^2 with real analytic boundary has a noncompact automorphism group, then it is biholomorphically equivalent to a domain \displaystyle E_m=\{\,z\in\mathbf{C}^2:\,\vert z_1\vert^{2m}+\vert z_2\vert^2
ABSTRACT We study the iteration of the family of maps given, by 3-step linear fractional recurren... more ABSTRACT We study the iteration of the family of maps given, by 3-step linear fractional recurrences. This family was studied earlier from the point of view of finding periodicities. In this paper we finish that study by determining all possible periods within this family. The novelty of our approach is that we apply the methods of complex dynamical systems. This leads to two classes of interesting pseudo automorphisms of infinite order. One of the classes consists of completely integrable maps. The other class consists of maps of positive entropy which have an invariant family of K3 surfaces.
Parabolic bifurcations in one complex dimension demonstrate a wide variety of interesting dynamic... more 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 $F_\epsilon: M\to M$ given locally by $$F_\epsilon(x,y) = (x + x^2 + \epsilon^2+ ..., b_\epsilon y+...)$$ where $|b_\epsilon|<1$, and the `$...$' terms involve $x$, $y$ and $\epsilon$.
We consider the family of quadratic H\'enon diffeomorphisms of the plane ${\bf R}^2$. A map w... more We consider the family of quadratic H\'enon 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 $p_c(z) = z^2$.
For any polynomial diffeomorphism $f$ of $\mathbb{C}^2$ with positive entropy, neither the Julia ... more 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.
We consider the real dynamics of a two parameter family of plane birational maps, focusing especi... more 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
Page 1. BOUNDARY CONTINUITY OF PROPER HOLOMORPHIC CORRESPONDENCES Eric Bedford Indiana University... more Page 1. BOUNDARY CONTINUITY OF PROPER HOLOMORPHIC CORRESPONDENCES Eric Bedford Indiana University Bloomington, Indiana 47405Steve Bell Princeton University Princeton, New Jersey 08544 i. Introduction ...
... for some neighborhood W of Page 4. 262 ERIC BEDFORD and JOHN ERIK FORN^SS z, Vz П W may be ob... more ... for some neighborhood W of Page 4. 262 ERIC BEDFORD and JOHN ERIK FORN^SS z, Vz П W may be obtained as the union of all real analytic curves in W starting at z whose tangents lie in H (oil). Since d il is real analytic ...
It is proved that proper holomorphic mappings between n-dimensional bounded pseudoconvex domains ... more It is proved that proper holomorphic mappings between n-dimensional bounded pseudoconvex domains with real analytic boundaries extend holomorphically past the boundary whenever the target domain is strictly pseudoconvex.
ABSTRACT It is shown that if a bounded domain \Omega\subset\mathbf{C}^2 with real analytic bounda... more ABSTRACT It is shown that if a bounded domain \Omega\subset\mathbf{C}^2 with real analytic boundary has a noncompact automorphism group, then it is biholomorphically equivalent to a domain \displaystyle E_m=\{\,z\in\mathbf{C}^2:\,\vert z_1\vert^{2m}+\vert z_2\vert^2
ABSTRACT We study the iteration of the family of maps given, by 3-step linear fractional recurren... more ABSTRACT We study the iteration of the family of maps given, by 3-step linear fractional recurrences. This family was studied earlier from the point of view of finding periodicities. In this paper we finish that study by determining all possible periods within this family. The novelty of our approach is that we apply the methods of complex dynamical systems. This leads to two classes of interesting pseudo automorphisms of infinite order. One of the classes consists of completely integrable maps. The other class consists of maps of positive entropy which have an invariant family of K3 surfaces.
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