On the Dynamics of the Family axd(x − 1) + x


Alzahra University, Vanak, Tehran, Iran


In this paper we consider the dynamics of the real polynomials of degree d + 1 with a fixed point of multiplicity d ≥ 2. Such polynomials are conjugate to fa,d(x) = axd(x−1)+x, a ∈ R\{0}, d ∈ N. Our aim is to study the dynamics fa,d in some special cases.


1. M. Akbari, M. Rabii, Real cubic polynomials with a fixed point of multiplicity two, to appear in Indagationes Mathematicae.
2. M. Akbari, M. Rabii, Hyperbolicity of the family fc(x) = c(x− x3/3 ), Iranian Journal of Mathematical Sciences and Informatics, Vol.6, No.1 , (2011), 53–58.
3. K. T. Alligood, T. D. Sauer, J. A. Yorke, Chaos, an introduction to dynamical systems, Springer-Verlag, (2000).
4. B. Branner, J. H. Hubbard, The iteration of cubic polynomials, Part I. The global topology of parameter space, Acta Mathematica, 160, (1988), 143–206.
5. W. de Melo, S. van Strien, One-dimensional dynamics, Springer-Verlag, (1993).
6. R. Devaney, An introduction to chaotic dynamical systems, 2nd. ed., Addison-Wesley, (1989).
7. S. N. Elaydi, Discrete chaos, with applications in science and engineering, 2nd. ed. Chapman and Hall/CRC, (2007).
8. J. Milnor, Cubic polynomial maps with periodic critical orbit, Part I. Complex Dynamics, Family and Friends . Ed. D. Schleicher. A. K. Peters Ltd, Wellesley, MA, (2009).
9. J. Milnor, Cubic polynomial maps with periodic critical orbit, Part II. Escape Regions, Stony Brook IMS Preprint, (2009/3).
10. J. Palis, A global view of dynamics and a conjecture of the denseness of finitude of attractors, Asterique, 261, (2000), 335–347.
11. P. Roesch, Cubic polynomials with a parabolic point, Ergod. Th. and Dynam. Sys. 30, (2010), 1843–1867.