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Julia-Fatou-Sullivan theory for real one-dimensional dynamics. (English) Zbl 0761.58007
Denoting by $$N$$ either the circle $$S'$$ or a compact interval of the real line, the authors study the dynamics of a smooth endomorphism $$f$$ of $$N$$ which is not a homeomorphism (if $$f$$ is a homeomorphism then it corresponds to a degree $$\pm1$$ rational map of the Riemann sphere). The singular set of $$f$$, $$\text{Sing}(f)$$, is defined as the union of the set of turning points of $$f$$ (a turning point is a critical noninflection point) and the boundary points of $$N$$ — in the case when $$f$$ has turning points — and as the set of fixed points of $$f$$ — when $$f$$ has no turning points. The Julia set of $$f$$, $$J(f)$$, is defined to be the limit set of $$\text{Sing}(f)$$ and it is forward invariant. Thus its complement, called Fatou set $$F(f)$$, is backward invariant $$(f^{- 1}(F(f))\subset F(f))$$.
The main result of the paper is the following: If $$f:N\to N$$ is a smooth map such that all its critical points are non-flat (i.e., some higher derivative is non-zero), then all the connected components of $$F(f)$$ are eventually periodic (i.e., eventually mapped into a periodic component of $$F(f)$$) and the number of periodic components of $$F(f)$$ is finite.

##### MSC:
 37E99 Low-dimensional dynamical systems 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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