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Perturbations of critical fixed points of analytic maps. (English) Zbl 0810.30005
Camacho, C. (ed.) et al., Complex analytic methods in dynamical systems. Proceedings of the congress held at Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil, January 1992. Paris: Société Mathématique de France, Astérisque. 222, 407-422 (1994).
The author proves several results on perturbations of analytic maps near critical fixed points. Theorem 1 says that if $$h$$ is a perturbation of a complex analytic function $$f$$ having a nondegenerate critical fixed point $$\theta$$ and if $$q$$ and $$\sigma$$ are corresponding fixed and critical points of $$h$$ then $$| h(\sigma)- p|\leq |\sigma- p|$$ provided $$\text{Re}((\theta- p)f''(\theta))< 0$$. Applying this theorem to $$f(z)= z^ d+ (d/(d- 1))z$$ and $$p= 0$$ the author derives a stronger version of Smale’s mean value conjecture for perturbations of this $$f$$.
For the entire collection see [Zbl 0797.00019].

##### MSC:
 30C10 Polynomials and rational functions of one complex variable 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
##### Keywords:
analytic maps; perturbation; critical fixed point
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