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Gibbs measures for hyperbolic attractors defined by densities

  • * Corresponding author: Mark Pollicott

    * Corresponding author: Mark Pollicott

The second author is partly supported by ERC-Advanced Grant 833802-Resonances and EPSRC grant EP/T001674/1.

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  • In this article we will describe a new construction for Gibbs measures for hyperbolic attractors generalizing the original construction of Sinai, Bowen and Ruelle of SRB measures. The classical construction of the SRB measure is based on pushing forward the normalized volume on a piece of unstable manifold. By modifying the density at each step appropriately we show that the resulting measure is a prescribed Gibbs measure. This contrasts with, and complements, the construction of Climenhaga-Pesin-Zelerowicz who replace the volume on the unstable manifold by a fixed reference measure. Moreover, the simplicity of our proof, which uses only explicit properties on the growth rate of unstable manifold and entropy estimates, has the additional advantage that it applies in more general settings.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Figure 1.  The push forward of the measure $ \lambda_n $ on $ W_\delta^u(x) $ by $ f^{k} $

    Figure 2.  A piece of local unstable manifold $ W^u_\delta(x) $ of length $ \ell $ for the Arnol'd CAT map and its image $ f^nW^u_\delta(x) $

    Figure 3.  When $ G = 0 $ the two top bar charts represent $ \mu_n(B_1) $ and $ \mu_n(B_2) $ for $ n = 1, \cdots, 12 $ and the convergence to $ \frac{\pi^2}{9} $ can be seen. (b) When $ G(x,y) = \frac{1}{10}\sin \left( 2\pi x \right) $ the two lower bar charts represent $ \mu_n(B_1) $ and $ \mu_n(B_2) $ for $ n = 1, \cdots, 12 $. The sequences $ \mu_n(B_1) $ and $ \mu_n(B_2) $ converge to the measures of the balls $ B_1 $ and $ B_2 $ with respect to the corresponding Gibbs measure. It its empirically clear these values are different, but the precise values are unknown to us

    Figure 4.  The push forward $ f^m W_\delta^u(x) $ is $ \epsilon $-dense

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