# On entropy of patterns given by interval maps

Fundamenta Mathematicae (1999)

- Volume: 162, Issue: 1, page 1-36
- ISSN: 0016-2736

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topBobok, Jozef. "On entropy of patterns given by interval maps." Fundamenta Mathematicae 162.1 (1999): 1-36. <http://eudml.org/doc/212410>.

@article{Bobok1999,

abstract = {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].},

author = {Bobok, Jozef},

journal = {Fundamenta Mathematicae},

keywords = {interval map; topological entropy; cycle; pattern},

language = {eng},

number = {1},

pages = {1-36},

title = {On entropy of patterns given by interval maps},

url = {http://eudml.org/doc/212410},

volume = {162},

year = {1999},

}

TY - JOUR

AU - Bobok, Jozef

TI - On entropy of patterns given by interval maps

JO - Fundamenta Mathematicae

PY - 1999

VL - 162

IS - 1

SP - 1

EP - 36

AB - 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].

LA - eng

KW - interval map; topological entropy; cycle; pattern

UR - http://eudml.org/doc/212410

ER -

## References

top- [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. Zbl0127.13102
- [2] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Adv. Ser. Nonlinear Dynam. 5, World Sci., Singapore, 1993. Zbl0843.58034
- [3] L. Alsedà, J. Llibre and M. Misiurewicz, Periodic orbits of maps of Y, Trans. Amer. Math. Soc. 313 (1989), 475-538. Zbl0803.54032
- [4] S. Baldwin, Generalizations of a theorem of Sharkovskii on orbits of continuous real-valued functions, Discrete Math. 67 (1987), 111-127. Zbl0632.06005
- [5] A. Berman and R. J. Plemmons, Non-Negative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. Zbl0484.15016
- [6] L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1992.
- [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. Zbl0447.58028
- [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. Zbl0842.58018
- [9] A. Blokh, Functional rotation numbers for one dimensional maps, Trans. Amer. Math. Soc. 347 (1995), 499-514.
- [10] A. Blokh, On rotation intervals for interval maps, Nonlinearity 7 (1994), 1395-1417. Zbl0809.58008
- [11] A. Blokh and M. Misiurewicz, Entropy of twist interval maps, Israel J. Math. 102 (1997), 61-99. Zbl0885.54016
- [12] J. Bobok and M. Kuchta, X-minimal patterns and generalization of Sharkovskii's theorem, Fund. Math. 156 (1998), 33-66. Zbl0909.26003
- [13] J. Bobok and I. Marek, On asymptotic behaviour of solutions of difference equations in partially ordered Banach spaces, submitted to Positivity.
- [14] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414. Zbl0212.29201
- [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. Zbl0725.58011
- [16] M. Denker, Ch. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Math. 527, Springer, 1976. Zbl0328.28008
- [17] T. Fort, Finite Difference and Difference Equations in the Real Domain, Oxford Univ. Press, 1965.
- [18] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. 54, Cambridge Univ. Press, 1995. Zbl0878.58020
- [19] M. Misiurewicz, Minor cycles for interval maps, Fund. Math. 145 (1994), 281-304. Zbl0818.58014
- [20] M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 456 (1990).
- [21] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63. Zbl0445.54007
- [22] W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122 (1966), 368-378. Zbl0146.18604
- [23] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1974.
- [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. Zbl0354.54027

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