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Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems

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  • We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. Under these conditions, we show that the coupled systems have invariant cones and possess SRB measures even though there are genuine obstructions to uniform hyperbolicity.

    Mathematics Subject Classification: Primary: 37C40; Secondary: 37D25, 37D30.


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  • Figure 1.  (Courtesy of Bastien Fernandez) Illustration of a coupled system having an attractor with wild geometry. The {plotted} points represent the intersection of the attractor $ \Lambda $ with a $ W^{cs}- $manifold through one of the points in the attractor. The horizontal and vertical axes indicate approximate directions of $ E^s $ and $ E^c $

    Figure 2.  Top: Evolution of a curve in $ W^{cu} $. Here we show the action of the map on a curve $ \gamma_0 $, $ \gamma_0(J) \subset \{z = \mbox{constant}\} $, such that $ \pi_{xy}\gamma_0 $ is contained in a $ W^u_A- $curve. The pictures on the top show the effect of $ \tilde f_1 $, $ \tilde f_2 $, and $ \tilde f_3 $ on $ \gamma_0 $, where $ \tilde f_1 $, $ \tilde f_2 $, and $ \tilde f_3 $ are the lifts of $ f_1 $, $ f_2 $, and $ f_3 $ respectively. Here we assume that $ r(x) = x $. Bottom: action of $ \tilde F^6 $ on a piece of curve $ \gamma_0 $, where $ \tilde F $ is the lift of the map $ F $. These plots highlight the effect of the monotonicity of the coupling

    Figure 3.  The pictures above illustrate assumptions (B2) and (B3). On the left is the graph of an example of $ r $: $ r = 0 $ outside of $ I_\epsilon $ and is very steep when $ r(x) $ is in between $ d $ and $ 1-d $. On the right is the graph of an example of $ g $. The highlighted intervals are $ g(I_+)\backslash I_+ $; each component has diameter less than $ d $

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