## Real Analysis Exchange

### Irrational Twist Systems for Interval Maps

#### Abstract

Let $I$ be a compact real interval and $f\colon ~I\rightarrow I$ continuous. We describe a special infinite minimal subsystem - we call it irrational twist system - of dynamical system $(I,f)$. We show that any twist system has an extremely regular behavior and it can be considered as an interval analogy of the irrational circle rotation.

#### Article information

Source
Real Anal. Exchange, Volume 27, Number 2 (2001), 441-456.

Dates
First available in Project Euclid: 2 June 2008

https://projecteuclid.org/euclid.rae/1212412847

Mathematical Reviews number (MathSciNet)
MR1922660

Zentralblatt MATH identifier
1096.37020

#### Citation

Bobok, Jozef; Kuchta, Milan. Irrational Twist Systems for Interval Maps. Real Anal. Exchange 27 (2001), no. 2, 441--456. https://projecteuclid.org/euclid.rae/1212412847

#### References

• L. S. Block, W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics 1513, Springer, Berlin, 1992.
• A. Blokh, Rotation numbers, twists and a Sharkovskii-Misiurewicz-type ordering for patterns on the interval, Ergodic Theory and Dynamical Systems, 15 (1995), 1331–1337.
• A. Blokh, Rotation Number for Unimodal Maps, MSRI (1994), Preprint, 58–94.
• A. Blokh, On Rotation Intervals for Interval Maps, Nonlinearity, 7 (1994), 1395–1417.
• A. Blokh, M. Misiurewicz, Entropy of Twist Interval maps, Israel Journal of Mathematics, 102 (1997), 61–99.
• J. Bobok, On entropy of patterns given by interval maps, Fundamenta Mathematicae, 162 (1999), 1–36.
• J. Bobok, On the topological entropy of green interval maps, Journal of Applied Analysis,7 (2001), 107–112.
• J. Bobok, Twist systems on the interval, Fundamenta Mathematicae, 19 pages, submitted.
• J. Bobok, M. Kuchta, Invariant measures for maps of the interval that do not have points of some period, Ergodic Theory and Dynamical Systems, 14 (1994), 9–21.
• J. Bobok, M. Kuchta, X–minimal patterns and a Generalization of Sharkovski\v\i's Theorem, Fundamenta Mathematicae, 156 (1998), 33–66.
• M. Denker, Ch. Grillenberger, K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Math., 527 Springer, Berlin, 1977.
• H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.
• R. R. Phelps, Lectures on Choquet's Theorem D. Van Nostrand Company, Inc., Princeton, New Jersey, 1966.