Jozef Bobok
580 California St., Suite 400
San Francisco, CA, 94104
On entropy of patterns given by interval maps
1999, Fundamenta Mathematicae
Sign up for access to the world's latest research
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].
References (26)
- R. L. A d l e r, A. G. K o n h e i m and M. H. M c A n d r e w, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319.
- L. A l s e d à, J. L l i b r e and M. M i s i u r e w i c z, Combinatorial Dynamics and Entropy in Dimension One, Adv. Ser. Nonlinear Dynam. 5, World Sci., Singapore, 1993.
- -, -, -, Periodic orbits of maps of Y , Trans. Amer. Math. Soc. 313 (1989), 475-538.
- S. B a l d w i n, Generalizations of a theorem of Sharkovskii on orbits of continuous real-valued functions, Discrete Math. 67 (1987), 111-127.
- A. B e r m a n and R. J. P l e m m o n s, Non-Negative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
- L. S. B l o c k and W. A. C o p p e l, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1992.
- L. B l o c k, J. G u c k e n h e i m e r, M. M i s i u r e w i c z and L. S. Y o u n g, 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.
- A. B l o k h, Rotation numbers, twists and a Sharkovskii-Misiurewicz-type order for patterns on the interval, Ergodic Theory Dynam. Systems 15 (1995), 1-14.
- -, Functional rotation numbers for one dimensional maps, Trans. Amer. Math. Soc. 347 (1995), 499-514.
- -, On rotation intervals for interval maps, Nonlinearity 7 (1994), 1395-1417.
- A. B l o k h and M. M i s i u r e w i c z, Entropy of twist interval maps, Israel J. Math. 102 (1997), 61-99.
- J. B o b o k and M. K u c h t a, X-minimal patterns and generalization of Sharkovskii's theorem, Fund. Math. 156 (1998), 33-66.
- J. B o b o k and I. M a r e k, On asymptotic behaviour of solutions of difference equa- tions in partially ordered Banach spaces, submitted to Positivity.
- R. B o w e n, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414.
- E. M. C o v e n and M. C. H i d a l g o, On the topological entropy of transitive maps of the interval , Bull. Austral. Math. Soc. 44 (1991), 207-213.
- M. D e n k e r, Ch. G r i l l e n b e r g e r and K. S i g m u n d, Ergodic Theory on Compact Spaces, Lecture Notes in Math. 527, Springer, 1976.
- T. F o r t, Finite Difference and Difference Equations in the Real Domain, Oxford Univ. Press, 1965.
- A. K a t o k and B. H a s s e l b l a t t, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. 54, Cambridge Univ. Press, 1995.
- M. M i s i u r e w i c z, Minor cycles for interval maps, Fund. Math. 145 (1994), 281- 304.
- M. M i s i u r e w i c z and Z. N i t e c k i, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 456 (1990).
- M. M i s i u r e w i c z and W. S z l e n k, Entropy of piecewise monotone mappings, Stu- dia Math. 67 (1980), 45-63.
- W. P a r r y, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122 (1966), 368-378.
- W. R u d i n, Real and Complex Analysis, McGraw-Hill, New York, 1974.
- P. Š t e f a n, A theorem of Sharkovskii on the coexistence of periodic orbits of contin- uous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), 237-248.
- KM FSv ČVUT Thákurova 7
- 29 Praha 6, Czech Republic E-mail: erastus@mbox.cesnet.cz Received 16 May 1996; in revised form 27 October 1997, 20 October 1998 and 13 March 1999