# Moscow Mathematical Journal

Volume 23, Issue 2, April–June 2023 pp. 271–282.

Accesses to Parabolic Fixed Point from its Immediate Basin

**Authors**:
Khudoyor Mamayusupov (1)

**Author institution:**(1) New Uzbekistan University, Mustakillik Avenue, 54, Mirzo Ulugbek, 100007, Tashkent, Uzbekistan;

National University of Uzbekistan, Faculty of Mathematics, University street, 9, Olmazor, 100174, Tashkent, Uzbekistan;

Romanovsky Institute of Mathematics, Uzbekistan Academy of Sciences, University street, 4, 100174, Tashkent, Uzbekistan

**Summary: **

Accesses to a parabolic fixed point from its immediate basin for certain rational functions were completely described. As a corollary, dynamics on immediate parabolic basins for rational Newton maps of entire functions have been obtained. It is proved that every immediate parabolic basin contains invariant accesses to the parabolic fixed point. Moreover, among these accesses there exists a unique dynamically defined access where dynamics are attracted towards the parabolic fixed point, whereas for other accesses, if there are any, the dynamics are repelled. A description of rational Newton maps in terms of the partial fraction decomposition of rational functions is obtained.

2020 Math. Subj. Class. 30D05, 37F10, 37F20.

**Keywords:**Access to parabolic fixed point, basin of attraction, Newton method, Blaschke product, inner function.

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