Defining the complexity of a green pattern exhibited by an interval map, we give the best bounds of the topological entropy of a pattern with a given complexity. Moreover, we show that the topological entropy attains its strict minimum on the set of patterns with fixed eccentricity m/n at a unimodal X-minimal case. Using a different method, the last result was independently proved in[11].
@article{bwmeta1.element.bwnjournal-article-fmv162i1p1bwm,
author = {Jozef Bobok},
title = {On entropy of patterns given by interval maps},
journal = {Fundamenta Mathematicae},
volume = {159},
year = {1999},
pages = {1-36},
zbl = {0938.54037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv162i1p1bwm}
}
Bobok, Jozef. On entropy of patterns given by interval maps. Fundamenta Mathematicae, Tome 159 (1999) pp. 1-36. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv162i1p1bwm/
[00000] [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. | Zbl 0127.13102
[00001] [2] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Adv. Ser. Nonlinear Dynam. 5, World Sci., Singapore, 1993. | Zbl 0843.58034
[00002] [3] L. Alsedà, J. Llibre and M. Misiurewicz, Periodic orbits of maps of Y, Trans. Amer. Math. Soc. 313 (1989), 475-538. | Zbl 0803.54032
[00003] [4] S. Baldwin, Generalizations of a theorem of Sharkovskii on orbits of continuous real-valued functions, Discrete Math. 67 (1987), 111-127. | Zbl 0632.06005
[00004] [5] A. Berman and R. J. Plemmons, Non-Negative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. | Zbl 0484.15016
[00005] [6] L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1992.
[00006] [7] L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one dimensional maps, in: Global Theory of Dynamical Systems, Lecture Notes in Math. 819, Springer, Berlin, 1980, 18-34. | Zbl 0447.58028
[00007] [8] A. Blokh, Rotation numbers, twists and a Sharkovskii-Misiurewicz-type order for patterns on the interval, Ergodic Theory Dynam. Systems 15 (1995), 1-14. | Zbl 0842.58018
[00008] [9] A. Blokh, Functional rotation numbers for one dimensional maps, Trans. Amer. Math. Soc. 347 (1995), 499-514.
[00009] [10] A. Blokh, On rotation intervals for interval maps, Nonlinearity 7 (1994), 1395-1417. | Zbl 0809.58008
[00010] [11] A. Blokh and M. Misiurewicz, Entropy of twist interval maps, Israel J. Math. 102 (1997), 61-99. | Zbl 0885.54016
[00011] [12] J. Bobok and M. Kuchta, X-minimal patterns and generalization of Sharkovskii's theorem, Fund. Math. 156 (1998), 33-66. | Zbl 0909.26003
[00012] [13] J. Bobok and I. Marek, On asymptotic behaviour of solutions of difference equations in partially ordered Banach spaces, submitted to Positivity.
[00013] [14] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414. | Zbl 0212.29201
[00014] [15] E. M. Coven and M. C. Hidalgo, On the topological entropy of transitive maps of the interval, Bull. Austral. Math. Soc. 44 (1991), 207-213. | Zbl 0725.58011
[00015] [16] M. Denker, Ch. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Math. 527, Springer, 1976. | Zbl 0328.28008
[00016] [17] T. Fort, Finite Difference and Difference Equations in the Real Domain, Oxford Univ. Press, 1965.
[00017] [18] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. 54, Cambridge Univ. Press, 1995. | Zbl 0878.58020
[00018] [19] M. Misiurewicz, Minor cycles for interval maps, Fund. Math. 145 (1994), 281-304. | Zbl 0818.58014
[00019] [20] M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 456 (1990).
[00020] [21] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63. | Zbl 0445.54007
[00021] [22] W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122 (1966), 368-378. | Zbl 0146.18604
[00022] [23] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1974.
[00023] [24] P. Štefan, A theorem of Sharkovskii on the coexistence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), 237-248. | Zbl 0354.54027