scholarly journals Periodic attractors of perturbed one-dimensional maps

2013 ◽  
Vol 33 (5) ◽  
pp. 1519-1541
Author(s):  
O. KOZLOVSKI
Keyword(s):  
Critical Points ◽  
Cross Ratio ◽  
One Dimensional ◽  
Periodic Attractors ◽  
Ratio Distortion ◽  
Distortion Estimates ◽  
One Dimensional Maps

AbstractIn this paper we investigate how many periodic attractors maps in a small neighbourhood of a given map can have. For this purpose we develop new tools which help to make uniform cross-ratio distortion estimates in a neighbourhood of a map with degenerate critical points.

ON THE DESTRUCTION OF THREE-DIMENSIONAL TORI

1996 ◽  
Vol 06 (07) ◽  
pp. 1319-1332 ◽  
Author(s):  
U. FEUDEL ◽  
M. A. SAFONOVA ◽  
J. KURTHS ◽  
V. S. ANISHCHENKO
Keyword(s):  
Critical Points ◽  
Three Dimensional ◽  
Box Dimension ◽  
Direct Transition ◽  
One Dimensional ◽  
Onset Of Chaos ◽  
One Dimensional Maps ◽  
Noninvertible Maps

We have studied the direct transition from three-frequency oscillations on T3 to chaos. The onset of chaos is analyzed where the torus loses its smoothness in the neighbourhood of a resonance. We have proposed two methods to reveal the destruction of the torus T3. The first method is based on the iteration of parts of the attractor to demonstrate stretching and folding. The second one is related to the evolution of the critical surfaces of noninvertible maps, which are generalizations of the critical points in one-dimensional maps. With both techniques we can show that the torus is indeed broken at the onset of chaos. An important advantage of these methods is that they can be easily applied even in cases where the expansion is much smaller than the contraction. Note that the estimation of the box dimension of such attractors yields integer values, because the broken structure of the attractor is too tiny to be detected by estimating the capacity.

scholarly journals STATISTICAL PROPERTIES OF ONE-DIMENSIONAL MAPS WITH CRITICAL POINTS AND SINGULARITIES

2006 ◽  
Vol 06 (04) ◽  
pp. 423-458 ◽  
Author(s):  
K. DÍAZ-ORDAZ ◽  
M. P. HOLLAND ◽  
S. LUZZATTO
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Invariant Probability Measure ◽  
Return Time ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Time Asymptotics ◽  
Exponential Decay Of Correlations ◽  
One Dimensional Maps ◽  
Absolutely Continuous Invariant

We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for Hölder observations.

scholarly journals Theory of Finite-Entanglement Scaling at One-Dimensional Quantum Critical Points

2009 ◽  
Vol 102 (25) ◽  
Author(s):  
Frank Pollmann ◽  
Subroto Mukerjee ◽  
Ari M. Turner ◽  
Joel E. Moore
Keyword(s):  
Critical Points ◽  
Quantum Critical Points ◽  
One Dimensional ◽  
Quantum Critical

scholarly journals On the finiteness of attractors for one-dimensional maps with discontinuities

2021 ◽  
Vol 389 ◽  
pp. 107891
Author(s):  
P. Brandão ◽  
J. Palis ◽  
V. Pinheiro
Keyword(s):  
One Dimensional ◽  
One Dimensional Maps

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.

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