Kneading theory

Toby Hall

doi:10.4249/scholarpedia.3956

Weighted kneading theory of one-dimensional maps with a hole

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s016117120430428x ◽

2004 ◽

Vol 2004 (38) ◽

pp. 2019-2038 ◽

Cited By ~ 3

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.

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Kneading theory and rotation intervals for a class of circle maps of degree one

Nonlinearity ◽

10.1088/0951-7715/3/2/008 ◽

1990 ◽

Vol 3 (2) ◽

pp. 413-452 ◽

Cited By ~ 7

Author(s):

L Alseda ◽

F Manosas

Keyword(s):

Circle Maps ◽

Kneading Theory

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Isentropic Real Cubic Maps

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403007552 ◽

2003 ◽

Vol 13 (07) ◽

pp. 1701-1709 ◽

Cited By ~ 8

Author(s):

Nuno Martins ◽

Ricardo Severino ◽

J. Sousa Ramos

Keyword(s):

Topological Entropy ◽

Level Set ◽

Topological Invariant ◽

Symbolic Order ◽

Kneading Theory ◽

Theory Framework

Given a family of bimodal maps on the interval, we need to consider a second topological invariant, other than the usual topological entropy, in order to classify it. With this work, we want to understand how to use this second invariant to distinguish bimodal maps with the same topological entropy and, in particular, how this second invariant changes within a given type of topological entropy level set. In order to do that, we use the kneading theory framework and introduce a symbolic product * between kneading invariants of maps from the same topological entropy level set, for which we show that the second invariant is preserved. Finally, we also show that the change of the second invariant follows closely the symbolic order between bimodal kneading sequences.

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COMBINATORICS OF THE KNEADING MAP

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127495001010 ◽

1995 ◽

Vol 05 (05) ◽

pp. 1339-1349 ◽

Cited By ~ 17

Author(s):

H. BRUIN

Keyword(s):

Topological Structure ◽

Limit Set ◽

Geometric Properties ◽

Unimodal Maps ◽

Omega Limit Set ◽

Kneading Theory

The kneading map and the Hofbauer tower are tools, developed by F. Hofbauer and G. Keller, to study unimodal maps and the kneading theory. In this paper we survey the geometric properties of these tools. Results concerning the topological structure of the critical omega-limit set are obtained.

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The anharmonic route to chaos: kneading theory

Nonlinearity ◽

10.1088/0951-7715/6/3/001 ◽

1993 ◽

Vol 6 (3) ◽

pp. 349-367 ◽

Cited By ~ 8

Author(s):

P Glendinning

Keyword(s):

Route To Chaos ◽

Kneading Theory

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DEVIL'S STAIRCASE OF GAP MAPS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403006376 ◽

2003 ◽

Vol 13 (01) ◽

pp. 115-122 ◽

Cited By ~ 1

Author(s):

JUNG-CHAO BAN ◽

CHENG-HSIUNG HSU ◽

SONG-SUN LIN

Keyword(s):

Topological Entropy ◽

Devil's Staircase ◽

Unimodal Maps ◽

One Dimensional ◽

Devil’S Staircase ◽

Staircase Structure ◽

Entropy Functions ◽

Kneading Theory

This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos.

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