A case of monotone ratio growth for quadratic-like mappings

• Autorzy: Pałuba W.
• Języki publikacji: EN

Abstrakty

EN

This is a study of the monotone (in parameter) behavior of the ratios of the consecutive intervals in the nested family of intervals delimited by the itinerary of a critical point. We consider a one-parameter power-law family of mappings of the form fa = -|x|^ alpha +a. Here we treat the dynamically simplest situation, before the critical point itself becomes strongly attracting ; this corresponds to the kneading sequence RRR ..., or-in the quadratic family-to the parameters c belongs to [-1,0] in the Mandelbrot set. We allow the exponent alpha to be an arbitrary real number greater than 1.

Słowa kluczowe

EN

quadratic-like of interval; post-critical orbit; Poincare lenght; nonlinearity; Poincare push

PL

długość Poincarego; nieliniowość

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2004
• Tom: Vol. 52, nr 4
• Strony: 381--393
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Pałuba W.

• Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland,

Bibliografia

• [1] S. Dawson, R. Galeeva, J. Milnor and C. Tresser, A monotonicity conjecture for Real cubic maps, in: Real and Complex Dynamical Systems, NATO Adv. Sci. Inst. Ser. C Mat. Phys. Sci. 464, Kluwer, Dordrecht, 1995, 165-187.
• [2] V. Dragan, A. Jones and P. Stacey, Repeated radicals and the real Fatou theorem, Austral. Math. Soc. Gaz. 29 (2002), 259-268.
• [3] J. Graczyk and G. Świątek, Induced expansion for quadratic polynomials, Ann. Sci. École Norm. Sup. 29 (1996), 399-482.
• [4] J. Milnor and W. Thurston, Iterated maps of the interval, in: Dynamical Systems, Lecture Notes in Math. 1342, Springer, Berlin, 1988, 465-563.
• [5] W. Pałuba, On conjugacies of infinitely renormalizable maps, Ph.D. thesis, City Univ. of New York, New York, 1992.
• [6] D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, in: AMS Centennial Publications, Vol. 2, AMS, Providence, RI, 1992, 417-466.
• [7] M. Tsujii, A simple proof for monotonicity of entropy in the quadratic family, Ergodic Theory Dynam. Systems 20 (2000), 925-933.