On backward attractors of interval maps*

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

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Periodic structure of alternating continuous interval maps

The Journal of Difference Equations and Applications ◽

10.1080/10236190600772515 ◽

2006 ◽

Vol 12 (8) ◽

pp. 847-858 ◽

Cited By ~ 23

Author(s):

Jose S. Cánovas ◽

Antonio Linero

Keyword(s):

Periodic Structure ◽

Interval Maps

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UNIMODAL INTERVAL MAPS OBTAINED FROM THE MODIFIED CHUA EQUATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493000246 ◽

1993 ◽

Vol 03 (02) ◽

pp. 323-332 ◽

Cited By ~ 4

Author(s):

MICHAŁ MISIUREWICZ

Keyword(s):

Periodic Orbit ◽

Critical Point ◽

Real Line ◽

Positive Measure ◽

Schwarzian Derivative ◽

Large Range ◽

Unimodal Maps ◽

Interval Maps ◽

Central Piece ◽

Piecewise Continuous

Following Brown [1992, 1993] we study maps of the real line into itself obtained from the modified Chua equations. We fix our attention on a one-parameter family of such maps, which seems to be typical. For a large range of parameters, invariant intervals exist. In such an invariant interval, the map is piecewise continuous, with most of pieces of continuity mapped in a monotone way onto the whole interval. However, on the central piece there is a critical point. This allows us to find sometimes a smaller invariant interval on which the map is unimodal. In such a way, we get one-parameter families of smooth unimodal maps, very similar to the well-known family of logistic maps x ↦ ax(1−x). We study more closely one of those and show that these maps have negative Schwarzian derivative. This implies the existence of at most one attracting periodic orbit. Moreover, there is a set of parameters of positive measure for which chaos occurs.

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"CONNECT-THE-DOTS" TREE MAPS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127495001101 ◽

1995 ◽

Vol 05 (05) ◽

pp. 1427-1431

Author(s):

LLUÍS ALSEDÀ ◽

JOHN GUASCHI ◽

JÉRÔME LOS ◽

FRANCESC MAÑOSAS ◽

PERE MUMBRÚ

Keyword(s):

Piecewise Monotone ◽

Interval Maps ◽

Work In Progress ◽

Monotone Map ◽

The Given

We announce the main results of work in progress on piecewise monotone models for patterns of tree maps. More precisely, we define a notion of pattern for tree maps, and given such a pattern, we construct a tree and a piecewise monotone map on this tree with the same pattern. This piecewise monotone model has the least entropy among all models exhibiting the given pattern and has "minimal dynamics". We also give a formula to compute this minimal entropy directly from the pattern. These results generalize the known results for interval maps and the results from Li & Ye [1993].

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