The growth of topological entropy for one dimensional maps

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.

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Computing the Topological Entropy of General One-Dimensional Maps

Transactions of the American Mathematical Society ◽

10.2307/2001614 ◽

1991 ◽

Vol 323 (1) ◽

pp. 39 ◽

Cited By ~ 4

Author(s):

P. Gora ◽

A. Boyarsky

Keyword(s):

Topological Entropy ◽

One Dimensional ◽

One Dimensional Maps

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TOPOLOGICAL ENTROPY FOR MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740100281x ◽

2001 ◽

Vol 11 (05) ◽

pp. 1443-1446 ◽

Cited By ~ 10

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.

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

Nonlinearity ◽

10.1088/0951-7715/21/11/005 ◽

2008 ◽

Vol 21 (11) ◽

pp. 2555-2567 ◽

Cited By ~ 11

Author(s):

Ming-Chia Li ◽

Ming-Jiea Lyu ◽

Piotr Zgliczyński

Keyword(s):

Topological Entropy ◽

One Dimensional ◽

One Dimensional Maps

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Entropy in dimension one

Frontiers in Complex Dynamics ◽

10.23943/princeton/9780691159294.003.0015 ◽

2014 ◽

Cited By ~ 2

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.

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