Jacobian elliptic Chebyshev rational maps

We introduce the Jacobian elliptic Chebyshev rational maps with the the equidistributivity property (or briefly EDP) and the semi-group property like the Chebyshev polynomial maps. Simple methods are also discussed for generating sequences of i.i.d. p-ary random variables based on the Jacobian elliptic Chebyshev rational maps. Furthermore, we give conjectures on correlational properties of a real-valued trajectory generated by the Jacobian elliptic Chebyshev rational maps.

[1]  高橋 陽一郎.  名著発掘 「Statistical Independence in Probability,Analysis and Number Theory--The Carus Mathematical Monographs,Number 12,The Mathematical Association of America」Marc Kac--Marc Kacの珠玉の1冊 , 2001 .

[2]  Michel Loève,et al.  Probability Theory I , 1977 .

[3]  P. Billingsley,et al.  Probability and Measure , 1980 .

[4]  T. Kohda,et al.  Statistics of chaotic binary sequences , 1997, IEEE Trans. Inf. Theory.

[5]  Tohru Kohda.  Sequences of I.I.D. Binary Random Variables Using Chaotic Dynamics , 1998, SETA.

[6]  J. L. Massey,et al.  An introduction to contemporary cryptology , 1988, Proc. IEEE.

[7]  Michael C. Mackey,et al.  Chaos, Fractals, and Noise , 1994 .

[8]  S. Katsura,et al.  Exactly solvable models showing chaotic behavior II , 1985 .

[9]  A. J. Lawrance,et al.  Curved Chaotic Map Time Series Models and Their Stochastic Reversals , 1998 .

[10]  Ken Umeno,et al.  METHOD OF CONSTRUCTING EXACTLY SOLVABLE CHAOS , 1996, chao-dyn/9610009.

[11]  T. W. Körner.  Exercises for Fourier analysis , 1993 .

[12]  T. Geisel,et al.  Statistical properties of chaos in Chebyshev maps , 1984 .

[13]  Alan F. Beardon,et al.  Iteration of Rational Functions , 1991 .

[14]  S. M. Ulam,et al.  On Combination of Stochastic and Deterministic Processes , 1947 .

[15]  Marvin K. Simon,et al.  Spread Spectrum Communications Handbook , 1994 .

[16]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[17]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[18]  S. Grossmann,et al.  Invariant Distributions and Stationary Correlation Functions of One-Dimensional Discrete Processes , 1977 .

[19]  Tohru Kohda,et al.  Correlational properties of Chebyshev chaotic sequences , 2000 .

[20]  Carl Ludwig Siegel,et al.  Topics in complex function theory , 1969 .

[21]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .