Renormalization on iterated cubic maps
Abstract
This communication will discuss the dynamics of iterated cubic maps from the real line to itself, and will describe the renormalization of the parameter space for such maps using methods of symbolic dynamics.
References
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14. M. Misiurewicz, W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63.
15. J. Ringland, M. Schell, A Genealogy and bifurcation skeleton for cycles of the iterated two-extremum map of the interval, SIAM J. Math. Anal. 22 (1991), 1354-1371.
16. J. Ringland, C. Tresser, A Genealogy for finite kneading sequences of bimodal maps on the interval, Trans. Amer. Math. Soc. 347 (1995), 164-181.
17. J. Rothschild, On the computation of topological entropy, Thesis, Cuny 1971.
18. H. Skjolding, B. Branner-Jorgensen, P. Christiansen, H. Jensen, Bifurcations in discrete dynamical systems with cubic maps, SIAM J. Apl. Math. 43 (1983), 520-534.
2. P. Collet, J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser 1980.
3. S.P. Dawson, R. Galeeva, J. Milnor, C. Tresser, A monotonicity conjecture for real cubic maps, Preprint: IMS 93-11.
4. B. Derrida, A. Gervois, Y. Pomeau, Iteration of endomorphisms on the real axis and representation of numbers, Ann. Inst. Henri Poincaré A XXIX (1978), 305-356.
5. J.P. Lampreia, J. Sousa Ramos, Symbolic dynamics of bimodal maps, Portugaliae Mathematica 54 (1997), 1-18.
6. J.P. Lampreia, J. Sousa Ramos, Computing the topological entropy of bimodal maps, Proceedings of European Conference on Iteration Theory (ECIT87). World Scientific, 1989.
7. J.P. Lampreia, R. Severino, J. Sousa Ramos, A product for bimodal Markov shifts, Proceedings of European Conference on Iteration Theory (ECIT96), Grazer Math. Ber. (to appear).
8. R.S. Mackay, C. Tresser, Some flesh on the skeleton: the bifurcations of bimodal maps, Physica D 27 (1987), 412-422.
9. W. de Melo, S. van Strein, One-Dimensional Dynamics, Springer-Verlag, 1993.
10. J. Milnor, Remarks on iterated cubic maps, Stony Brook I.M.S. Preprint 1990 #6.
11. J. Milnor, W. Thurston, On iterated maps of the interval, Ed. J.C. Alexander. Proceedings Univ. Maryland 1986-1987. Lect. Notes in Math. 1342, Springer-Verlag, 1988, 465-563.
12. J. Milnor, C. Tresser, On Entropy and Monotonicity for Real Cubic Maps, Stony Brook I.M.S. Preprint 1998 #9.
13. C. Mira, Chaotic Dynamics, World Scientific, 1987.
14. M. Misiurewicz, W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63.
15. J. Ringland, M. Schell, A Genealogy and bifurcation skeleton for cycles of the iterated two-extremum map of the interval, SIAM J. Math. Anal. 22 (1991), 1354-1371.
16. J. Ringland, C. Tresser, A Genealogy for finite kneading sequences of bimodal maps on the interval, Trans. Amer. Math. Soc. 347 (1995), 164-181.
17. J. Rothschild, On the computation of topological entropy, Thesis, Cuny 1971.
18. H. Skjolding, B. Branner-Jorgensen, P. Christiansen, H. Jensen, Bifurcations in discrete dynamical systems with cubic maps, SIAM J. Apl. Math. 43 (1983), 520-534.
LampreiaJ. P., SeverinoR. J. M., & de Sousa RamosJ. R. S. (1999). Renormalization on iterated cubic maps. Annales Mathematicae Silesianae, 13, 193-204. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14148
José Paulo Lampreia
Departamento de Matemática Universidade Nova de Lisboa, Portugal Portugal
Departamento de Matemática Universidade Nova de Lisboa, Portugal Portugal
Ricardo José Mendes Severino
ricardo@math.uminho.pt
Departamento de Matemática, Universidade do Minho, Portugal Portugal
Departamento de Matemática, Universidade do Minho, Portugal Portugal
José Rodrigues Santos de Sousa Ramos
Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Portugal Portugal
Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Portugal Portugal
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