On the fractal structure of the rescaled evolution set of Carlitz sequences of polynomials. (English) Zbl 0958.68110
Summary: Self-similarity properties of the coefficient patterns of the so-called \(m\)-Carlitz sequences of polynomials are considered. These properties are coded in an associated fractal set – the rescaled evolution set. We extend previous results on linear cellular automata with states in a finite field. Applications are given for the sequence of Legendre polynomials and sequences associated with the zero Bessel function.
MSC:
| 68Q80 | Cellular automata (computational aspects) |
References:
| [1] | Allouche, J.-P.; Haeseler, F.v.; Peitgen, H.-O.; Skordev, G., Linear cellular automata, finite automata and Pascal’s triangle, Discrete Appl. Math., 66, 1-22 (1996) · Zbl 0854.68065 |
| [2] | J.-P. Allouche, G. Skordev, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Appl. Math., to appear.; J.-P. Allouche, G. Skordev, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Appl. Math., to appear. · Zbl 0976.33005 |
| [3] | J.-P. Allouche, M. Mendes France, Automata and automatic sequences, in: F. Axel et al. (Eds.), Beyond Quasicristals, Les Editions de Physique, Springer, Berlin, 1994, pp. 293-367.; J.-P. Allouche, M. Mendes France, Automata and automatic sequences, in: F. Axel et al. (Eds.), Beyond Quasicristals, Les Editions de Physique, Springer, Berlin, 1994, pp. 293-367. · Zbl 0881.11026 |
| [4] | Bandt, Ch., Self-similar sets I. Topological markov chains and mixed self-similar sets, Math. Nachr., 142, 107-123 (1989) · Zbl 0707.28004 |
| [5] | B. Bondarenko, Generalized Triangles and Pyramids of Pascal, Their Fractals, Graphs and Applications, Tashkend, Fan, 1990 (in Russian).; B. Bondarenko, Generalized Triangles and Pyramids of Pascal, Their Fractals, Graphs and Applications, Tashkend, Fan, 1990 (in Russian). · Zbl 0706.05002 |
| [6] | Carlitz, L., The coefficients of reciprocal of \(J_0(X)\), Arch. Math., 6, 121-127 (1955) · Zbl 0064.06502 |
| [7] | G. Edgar, Measure, Topology and Fractal Geometry, Springer, New York, 1990.; G. Edgar, Measure, Topology and Fractal Geometry, Springer, New York, 1990. · Zbl 0727.28003 |
| [8] | K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, New York, 1985.; K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, New York, 1985. · Zbl 0587.28004 |
| [9] | F.v. Haeseler, H.-O. Peitgen, G. Skordev, On the fractal structure of rescaled evolution sets of cellular automata and attractors of dynamical systems, Inst. Dyn. Syst., Univ. Bremen, Report 278, 1992.; F.v. Haeseler, H.-O. Peitgen, G. Skordev, On the fractal structure of rescaled evolution sets of cellular automata and attractors of dynamical systems, Inst. Dyn. Syst., Univ. Bremen, Report 278, 1992. · Zbl 1070.37006 |
| [10] | Haeseler, F.v.; Peitgen, H.-O.; Skordev, G., Pascal triangle, dynamical systems and attractors, Ergodic Theory Dynamical System, 12, 479-486 (1992) · Zbl 0784.58032 |
| [11] | Haeseler, F.v.; Peitgen, H.-O.; Skordev, G., Cellular automata, matrix substitutions and fractals, Ann. Math. Art. Intell., 8, 345-362 (1993) · Zbl 0866.68069 |
| [12] | Haeseler, F.v.; Peitgen, H.-O.; Skordev, G., Multifractal decomposition of rescaled evolution sets of equivariant cellular automata, Random Comput. Dynamics, 3, 93-119 (1995) · Zbl 0843.58076 |
| [13] | Haeseler, F.v.; Peitgen, H.-O.; Skordev, G., Global analysis of self-similarity features of cellular automata: selected examples, Physica D, 86, 64-80 (1995) · Zbl 0890.58030 |
| [14] | Lange, E.; Peitgen, H.-O.; Skordev, G., Fractal patterns in Gaussian and Stirling number tables, Ars Combin., 48, 3-26 (1998) · Zbl 0963.11014 |
| [15] | N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Inc., Coll. Publ., Englewood Cliffs, NJ, 1965.; N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Inc., Coll. Publ., Englewood Cliffs, NJ, 1965. · Zbl 0131.07002 |
| [16] | Mauldin, R.; Williams, S., Hausdorff dimension in graph directed construction, Trans. Amer. Math. Soc., 309, 811-829 (1989) · Zbl 0706.28007 |
| [17] | McIntosh, R., A generalization of a congruential property of Lucas, Amer. Math. Monthly, 99, 3, 231-238 (1992) · Zbl 0755.11001 |
| [18] | H.-O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals, Springer, New York, 1992.; H.-O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals, Springer, New York, 1992. · Zbl 0779.58004 |
| [19] | Salon, O., Suites automatiques á multi-indices et algébricité, C. R. Acad. Sci. Paris, Sér. I, 305, 501-504 (1987) · Zbl 0628.10007 |
| [20] | Sved, M.; Pitman, J., Divisibility of binomial coefficients by prime powers a geometrical approach, Ars Combin., 26A, 197-222 (1988) · Zbl 0673.10011 |
| [21] | Takahashi, S., Self-similarity of linear cellular automata, J. Comput. Sci., 44, 114-140 (1992) · Zbl 0743.68105 |
| [22] | Wahab, J., New cases of irreducibility for Legendre polynomials, Duke Math. J., 19, 165-176 (1952) · Zbl 0049.29602 |
| [23] | Willson, S., Cellular automata can generate fractals, Discr. Appl. Math., 8, 91-99 (1984) · Zbl 0533.68051 |
| [24] | Willson, S., Calculating growth rates and moments for additive cellular automata, Discrete Appl. Math., 35, 47-65 (1992) · Zbl 0738.68060 |
| [25] | S. Wolfram, Theory and Application of Cellular Automata, World Scientific Publ. Co. Pte. Lid., Singapore, 1996.; S. Wolfram, Theory and Application of Cellular Automata, World Scientific Publ. Co. Pte. Lid., Singapore, 1996. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
