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Real bounds, ergodicity and negative Schwarzian for multimodal maps
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Strien, Sebastian van and Vargas, Edson. (2004) Real bounds, ergodicity and negative Schwarzian for multimodal maps. American Mathematical Society. Journal, Vol.17 (No.4). pp. 749782. ISSN 08940347

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Official URL: http://dx.doi.org/10.1090/S0894034704004631
Abstract
Over the last 20 years, many of the most spectacular results in the field of
dynamical systems dealt specifically with interval and circle maps (or perturbations
and complex extensions of such maps). Primarily, this is because in the
onedimensional case, much better distortion control can be obtained than for general
dynamical systems. However, many of these spectacular results were obtained
so far only for unimodal maps. The aim of this paper is to provide all the tools for
studying general multimodal maps of an interval or a circle, by obtaining
* real bounds controlling the geometry of domains of certain first return maps,
and providing a new (and we believe much simpler) proof of absense of
wandering intervals;
* provided certain combinatorial conditions are satisfied, large real bounds
implying that certain first return maps are almost linear;
* Koebe distortion controlling the distortion of high iterates of the map, and
negative Schwarzian derivative for certain return maps (showing that the
usual assumption of negative Schwarzian derivative is unnecessary);
* control of distortion of certain first return maps;
* ergodic properties such as sharp bounds for the number of ergodic components.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics  
Divisions:  Faculty of Science > Mathematics  
Library of Congress Subject Headings (LCSH):  Mappings (Mathematics), Dynamics  
Journal or Publication Title:  American Mathematical Society. Journal  
Publisher:  American Mathematical Society  
ISSN:  08940347  
Official Date:  2004  
Dates: 


Volume:  Vol.17  
Number:  No.4  
Page Range:  pp. 749782  
Identification Number:  10.1090/S0894034704004631  
Status:  Peer Reviewed  
Access rights to Published version:  Open Access  
References:  [1] A. M. Blokh and M. Yu Lyubich, Measure and dimension of solenoidal attractors of onedimensional 

URI:  http://wrap.warwick.ac.uk/id/eprint/4300 
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