TOPOLOGICAL ENTROPY FOR MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS

2001 ◽  
Vol 11 (05) ◽  
pp. 1443-1446 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
PIOTR ZGLICZYŃSKI
Keyword(s):  
Topological Entropy ◽  
Positive Entropy ◽  
One Dimensional ◽  
Interval Map ◽  
One Dimensional Maps

We prove that if an interval map of positive entropy is perturbed to a compact multidimensional map then the topological entropy cannot drop down considerably if the perturbation is small.

scholarly journals Weighted kneading theory of one-dimensional maps with a hole

2004 ◽  
Vol 2004 (38) ◽  
pp. 2019-2038 ◽  
Author(s):  
J. Leonel Rocha ◽  
J. Sousa Ramos
Keyword(s):  
Hausdorff Dimension ◽  
Power Series ◽  
Formal Power Series ◽  
Topological Entropy ◽  
Escape Rate ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Formal Power ◽  
Kneading Theory

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

One-Dimensional Semiconjugacy Revisited

2003 ◽  
Vol 13 (07) ◽  
pp. 1657-1663 ◽  
Author(s):  
J. F. Alves ◽  
J. Sousa Ramos
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Linear Map ◽  
One Dimensional ◽  
Piecewise Linear Map ◽  
Piecewise Monotone ◽  
Positive Topological Entropy ◽  
Interval Map

Let f be a piecewise monotone interval map with positive topological entropy h(f)= log (s). Milnor and Thurston showed that f is topological semiconjugated to a piecewise linear map having slope s. Here we prove that these semiconjugacies are the eigenvectors of a certain linear endomorphism associated to f. Using this characterization, we prove a conjecture presented by those authors.

Topological entropy for multidimensional perturbations of snap-back repellers and one-dimensional maps

Nonlinearity ◽  
2008 ◽  
Vol 21 (11) ◽  
pp. 2555-2567 ◽  
Author(s):  
Ming-Chia Li ◽  
Ming-Jiea Lyu ◽  
Piotr Zgliczyński
Keyword(s):  
Topological Entropy ◽  
One Dimensional ◽  
One Dimensional Maps

Entropy in dimension one

2014 ◽  
Author(s):  
William P. Thurston
Keyword(s):  
Topological Entropy ◽  
Finite Interval ◽  
Algebraic Integer ◽  
Small Letter ◽  
One Dimensional ◽  
Original Manuscript ◽  
Postcritically Finite ◽  
Perron Number ◽  
One Dimensional Maps ◽  
Dimension One

This chapter studies the topological entropy h of postcritically finite one-dimensional maps and, in particular, the relations between dynamics and arithmetics of eʰ, presenting some constructions for maps with given entropy and characterizing what values of entropy can occur for postcritically finite maps. In particular, the chapter proves: h is the topological entropy of a postcritically finite interval map if and only if h = log λ‎, where λ‎ ≥ 1 is a weak Perron number, i.e., it is an algebraic integer, and λ‎ ≥ ∣λ‎superscript Greek Small Letter Sigma∣ for every Galois conjugate λ‎superscript Greek Small Letter Sigma ∈ C. Unfortunately, the author of this chapter has died before completing this work, hence this chapter contains both the original manuscript as well as a number of notes which clarify many of the points mentioned therein.

Periodic points and topological entropy of one dimensional maps

1980 ◽  
pp. 18-34 ◽  
Author(s):  
Louis Block ◽  
John Guckenheimer ◽  
Michal Misiurewicz ◽  
Lai Sang Young
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
One Dimensional ◽  
One Dimensional Maps
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