Computing the topological entropy of general one-dimensional maps

Góra, P.; Boyarsky, A.

doi:10.1090/S0002-9947-1991-1062871-7

Computing the topological entropy of general one-dimensional maps
HTML articles powered by AMS MathViewer

by P. Góra and A. Boyarsky PDF

Trans. Amer. Math. Soc. 323 (1991), 39-49 Request permission

Abstract:

A matrix-theoretic method for computing the topological entropy of continuous, piecewise monotonic maps of the interval is presented. The method results in a constructive procedure which is easily implemented on the computer. Examples for families of unimodal, nonunimodal and discontinuous maps are presented.

References

P. Collet, J. P. Crutchfield, and J.-P. Eckmann, Computing the topological entropy of maps, Comm. Math. Phys. 88 (1983), no. 2, 257–262. MR 696807, DOI 10.1007/BF01209479
Louis Block, James Keesling, Shi Hai Li, and Kevin Peterson, An improved algorithm for computing topological entropy, J. Statist. Phys. 55 (1989), no. 5-6, 929–939. MR 1002478, DOI 10.1007/BF01041072
M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), no. 1, 45–63. MR 579440, DOI 10.4064/sm-67-1-45-63
John Milnor and William Thurston, On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 465–563. MR 970571, DOI 10.1007/BFb0082847
R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 175106, DOI 10.1090/S0002-9947-1965-0175106-9

Periodic orbits and topological entropy of one-dimensional maps

T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc. 3 (1971), 176–180. MR 289746, DOI 10.1112/blms/3.2.176
Abraham Boyarsky, A matrix method for estimating the Liapunov exponent of one-dimensional systems, J. Statist. Phys. 50 (1988), no. 1-2, 213–229. MR 939487, DOI 10.1007/BF01022993
P. Góra, A. Boyarsky, and H. Proppe, Constructive approximations to densities invariant under nonexpanding transformations, J. Statist. Phys. 51 (1988), no. 1-2, 179–194. MR 952751, DOI 10.1007/BF01015325
H. Proppe, W. Byers, and A. Boyarsky, Singularity of topological conjugacies between certain unimodal maps of the interval, Israel J. Math. 44 (1983), no. 4, 277–288. MR 710233, DOI 10.1007/BF02761987
P. Góra and A. Boyarsky, Why computers like Lebesgue measure, Comput. Math. Appl. 16 (1988), no. 4, 321–329. MR 959419, DOI 10.1016/0898-1221(88)90148-4
Lai-Sang Young, Entropy, Lyapunov exponents, and Hausdorff dimension in differentiable dynamical systems, IEEE Trans. Circuits and Systems 30 (1983), no. 8, 599–607. MR 715515, DOI 10.1109/TCS.1983.1085395
Louis Block and Ethan M. Coven, Approximating entropy of maps of the interval, Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 237–241. MR 1102718
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
MichałMisiurewicz, Jumps of entropy in one dimension, Fund. Math. 132 (1989), no. 3, 215–226. MR 1002409, DOI 10.4064/fm-132-3-215-226