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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 . 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 ∈ [-1,0] in the Mandelbrot set. We allow the exponent α to be an arbitrary real number greater than 1.
Waldemar Pałuba. "A Case of Monotone Ratio Growth for Quadratic-Like Mappings." Bulletin of the Polish Academy of Sciences. Mathematics 52.4 (2004): 381-393. <http://eudml.org/doc/280522>.
@article{WaldemarPałuba2004, abstract = {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 $f_a = -|x|^\{α\} + a$. 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 ∈ [-1,0] in the Mandelbrot set. We allow the exponent α to be an arbitrary real number greater than 1.}, author = {Waldemar Pałuba}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, keywords = {quadratic-like mapping of an interval; post-critical orbit; Poincaré length; nonlinearity; Poincaré push}, language = {eng}, number = {4}, pages = {381-393}, title = {A Case of Monotone Ratio Growth for Quadratic-Like Mappings}, url = {http://eudml.org/doc/280522}, volume = {52}, year = {2004}, }
TY - JOUR AU - Waldemar Pałuba TI - A Case of Monotone Ratio Growth for Quadratic-Like Mappings JO - Bulletin of the Polish Academy of Sciences. Mathematics PY - 2004 VL - 52 IS - 4 SP - 381 EP - 393 AB - 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 $f_a = -|x|^{α} + a$. 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 ∈ [-1,0] in the Mandelbrot set. We allow the exponent α to be an arbitrary real number greater than 1. LA - eng KW - quadratic-like mapping of an interval; post-critical orbit; Poincaré length; nonlinearity; Poincaré push UR - http://eudml.org/doc/280522 ER -