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Eric Bedford

Publications and source records attributed to Eric Bedford.

40 records · Page 3Linked to original sources
arXiv1996

Polynomial diffeomorphisms of C^2: V. Critical points and Lyapunov exponents

In this paper we continue our study of polynomial diffeomorphisms of C^2. Let us recall that there is an invariant measure $μ$, which is the pluri-complex version of the harmonic measure of the Julia set for polynomial maps of C. In this paper we give an integral formula for the Lyapunov exponents of a polynomial automorphism with respect to $μ$ analogous to the Brolin-Manning formula polynomial maps of C. Our formula relates the Lyapunov exponents to the value of a Green function at a type of critical point which we define in this paper. We show that these critical points have a very natural dynamical interpretation.

math.CV↗
arXiv1993

Distribution of periodic points of polynomial diffeomorphisms of C^2

This paper deals with the dynamics of a simple family of holomorphic diffeomorphisms of $\C^2$: the polynomial automorphisms. This family of maps has been studied by a number of authors. We refer to [BLS] for a general introduction to this class of dynamical systems. An interesting object from the point of view of potential theory is the equilibrium measure $μ$ of the set $K$ of points with bounded orbits. In [BLS] $μ$ is also characterized dynamically as the unique measure of maximal entropy. Thus $μ$ is also an equilibrium measure from the point of view of the thermodynamical formalism. In the present paper we give another dynamical interpretation of $μ$ as the limit distribution of the periodic points of $f$.

math.DS↗
arXiv1992

Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $μ^\pm$ and the equilibrium measure $μ=μ^+\wedgeμ^-$. In this paper we study some geometric and dynamical properties of these objects. First, we characterize $μ$ as the unique measure of maximal entropy. Then we show that the measure $μ$ has a local product structure and that the currents $μ^\pm$ have a laminar structure. This allows us to deduce information about periodic points and heteroclinic intersections. For example, we prove that the support of $μ$ coincides with the closure of the set of saddle points. The methods used combine the pluripotential theory with the theory of non-uniformly hyperbolic dynamical systems.

math.DS↗