Recurrence analysis on Julia sets of semigroups of complex polynomials

Abstract

We introduce a recurrence function in order to analyze the dynamics of semigroups of complex polynomials. We show that under a regularity hypothesis, the recurrence function is continuous in the complex plane. This is a new notion even for the case of a semigroup with just one generator.

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Acknowledgements

The authors would like to thank Mr.Arnold Ramírez for developing the code of some of the software needed for the experiments. The authors would also like to thank the anonymous referees for their useful suggestions and for pointing out several important references.

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Correspondence to Renato Colucci.

Additional information

This work was partially supported by the Project 5533 of the Department of Mathematics, Pontificia Universidad Javeriana.

The third author was supported by Austrian Science Fund project FWF P24028-N18.

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Chacón, G.R., Colucci, R. & D’Angeli, D. Recurrence analysis on Julia sets of semigroups of complex polynomials. J. Appl. Math. Comput. 46, 201–214 (2014). https://doi.org/10.1007/s12190-013-0746-1

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Keywords

  • Time series analysis
  • Semigroups of analytic functions
  • Julia set
  • Complex dynamics

Mathematics Subject Classification (2000)

  • 37M10
  • 37F50