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Forts of quadratic polynomials under iteration

2017; Elsevier BV; Volume: 331; Linguagem: Inglês

10.1016/j.cam.2017.09.008

1879-1778

Zhiheng Yu, Weinian Zhang,

Mathematical Dynamics and Fractals

Since some dynamical behaviors of a one-dimensional mapping are influenced by the number of forts, attention is paid to the change of the number under iteration. For simple computation we work on quadratic polynomials. We use the theory of polynomial complete discrimination system to give a symbolic algorithm for the number of forts of iterated polynomials and apply the algorithm to quadratic functions, which proves an alternative result that the number either persists to be 1 or tends to infinity under iteration. We further compute the number for iterates of order 2,3,…,7 in the above infinity case and obtain critical values of the parameter at which the number changes. Those changes with finitely many data display a conjectured Fibonacci rule.