Nonlinearity, Volume (31), No (11), Year (2018-10) , Pages (5329-5349)

Title : ( Invariant graphs for chaotically driven maps )

Authors: sara fadaei , G. Keller , Fateme Helen Ghane Ostadghassemi ,

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‎This paper investigates the geometrical structures of invariant graphs of skew product systems of the form $F‎ : ‎\Theta \times \I \to \Theta \times \I‎ , ‎(\theta,y)\mapsto (S\theta,\f(y))$ driven by a hyperbolic base map $S‎ : ‎\Theta \to \Theta$‎ ‎(e.g‎. ‎a baker map or an Anosov surface diffeomorphism)‎ ‎and with monotone increasing fibre maps $(f_{\theta})_{\theta \in \Theta}$ having negative Schwarzian derivatives‎. ‎We recall a classification‎, ‎with respect to the number and to the Lyapunov exponents of invariant graphs‎, ‎for this class of systems‎. ‎Our major goal here is to describe the structure of invariant graphs and study the properties of the‎ ‎pinching set‎, ‎the set of points where the values of all of the invariant graphs coincide‎. ‎In \cite{KO2}‎, ‎the authors studied skew product systems driven by a generalized baker map $S:\Tt\to\Tt$ with the restrictive assumption that $\f$ depend on $\theta=\2$ only through the stable coordinate $x$ of $\theta$‎. ‎Our aim is to relax this assumption and construct a fibre-wise conjugation‎ ‎between the original system and a new system for which the fibre maps depend only on the stable coordinate of the‎ drive. As an application of this construction we prove that, when S is an Anosov diffeomorphism, a pinching set is a union of global unstable fibers for S. ‎


, Invariant graph‎, ‎skew product‎, ‎synchronization‎, ‎negative Schwarzian derivative‎, ‎pinch‎ ‎points.
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author = {Fadaei, Sara and G. Keller and Ghane Ostadghassemi, Fateme Helen},
title = {Invariant graphs for chaotically driven maps},
journal = {Nonlinearity},
year = {2018},
volume = {31},
number = {11},
month = {October},
issn = {0951-7715},
pages = {5329--5349},
numpages = {20},
keywords = {Invariant graph‎; ‎skew product‎; ‎synchronization‎; ‎negative Schwarzian derivative‎; ‎pinch‎ ‎points.},


%0 Journal Article
%T Invariant graphs for chaotically driven maps
%A Fadaei, Sara
%A G. Keller
%A Ghane Ostadghassemi, Fateme Helen
%J Nonlinearity
%@ 0951-7715
%D 2018