Bifurcation currents in holomorphic dynamics on
-
Giovanni Bassanelli
and
François Berteloot
Abstract
We use pluri-potential theory to study the bifurcations of holomorphic families {ƒλ}λ∈X of rational maps on 
or endomorphisms of 
. To this purpose we establish some formulas for L(ƒλ) and ddcL(ƒλ) where L(ƒλ) is the sum of the Lyapunov exponents of ƒλ with respect to the maximal entropy measure. We show that the bifurcation current ddcL(ƒλ) both detects the instability of repulsive cycles and the interaction between critical and Julia sets.
For families of rational maps of degree d, we introduce a bifurcation measure defined by (ddcL(ƒλ))2d−2 add study its first properties. In particular, we show that the support of this measure is contained in the closure of the set of rational maps having 2d − 2 distinct Cremer-Cycles. This approach yields to a purely potential-theoretic proof of Mañé-Sad-Sullivan theorem and, moreover, allows us to extend it.
© Walter de Gruyter
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- Tangent spaces and Gromov-Hausdorff limits of subanalytic spaces
- Pinching estimates and motion of hypersurfaces by curvature functions
- Characters, supercharacters and Weber modular functions
- Three-dimensional Ricci solitons which project to surfaces
- Some q-analogues of the Carter-Payne theorem
- Preperiodic points of polynomials over global fields
- Orbit-counting in non-hyperbolic dynamical systems
- On intervals with few prime numbers
-
Bifurcation currents in holomorphic dynamics on
Articles in the same Issue
- Tangent spaces and Gromov-Hausdorff limits of subanalytic spaces
- Pinching estimates and motion of hypersurfaces by curvature functions
- Characters, supercharacters and Weber modular functions
- Three-dimensional Ricci solitons which project to surfaces
- Some q-analogues of the Carter-Payne theorem
- Preperiodic points of polynomials over global fields
- Orbit-counting in non-hyperbolic dynamical systems
- On intervals with few prime numbers
-
Bifurcation currents in holomorphic dynamics on
