Dynamics of Rational Maps: a Current on the Bifurcation Locus
نویسنده
چکیده
Let fλ : P 1 → P be a family of rational maps of degree d > 1, parametrized holomorphically by λ in a complex manifold X. We show that there exists a canonical closed, positive (1,1)-current T on X supported exactly on the bifurcation locus B(f) ⊂ X. If X is a Stein manifold, then the stable regime X − B(f) is also Stein. In particular, each stable component in the space Polyd (or Ratd) of all polynomials (or rational maps) of degree d is a domain of holomorphy.
منابع مشابه
Dynamics of Rational Maps: Lyapunov Exponents, Bifurcations, and Capacity
Let L(f) = ∫ log ‖Df‖ dμf denote the Lyapunov exponent of a rational map, f : P → P. In this paper, we show that for any holomorphic family of rational maps {fλ : λ ∈ X} of degree d > 1, T (f) = ddL(fλ) defines a natural, positive (1,1)-current on X supported exactly on the bifurcation locus of the family. The proof is based on the following potential-theoretic formula for the Lyapunov exponent...
متن کاملMisiurewicz Parameters and Dynamical Stability of Polynomial-like Maps of Large Topological Degree
The goal of this paper is to study the dynamical stability of polynomial-like maps of large topological degree, generalizing to this setting the theory developed in [BBD15] for endomorphisms of P . Our main result relates bifurcation in such families with the volume growth of the postcritical set under iteration, generalizing the one-dimensional equivalence between dynamical stability and norma...
متن کاملThe Mandelbrot set is universal
We show small Mandelbrot sets are dense in the bifurcation locus for any holomorphic family of rational maps.
متن کاملOn the geometry of bifurcation currents for quadratic rational maps
We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive (1, 1)-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.
متن کاملNonlinear Dynamics and Control of Crank-Slider Mechanism with Multiple Clearance Joints
In the current study, behavior of crank-slider mechanism with single and multiple clearance joints are analyzed. Using Lankarani-Nikravesh theory for estimating discontinuous contact forces in clearance joints, relevant systems have been mathematically modeled. Through numerical simulations, perturbations in response of mechanisms with clearance joints have been analyzed. Effects of increasing ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2001