# American Institute of Mathematical Sciences

August  2022, 42(8): 3953-3977. doi: 10.3934/dcds.2022038

## Gibbs measures for hyperbolic attractors defined by densities

 Department of Mathematics, Warwick University, Coventry, CV4 7AL, United Kingdom

* Corresponding author: Mark Pollicott

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

Received  September 2021 Revised  February 2022 Published  August 2022 Early access  April 2022

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.

Citation: David Parmenter, Mark Pollicott. Gibbs measures for hyperbolic attractors defined by densities. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3953-3977. doi: 10.3934/dcds.2022038
##### References:
 [1] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Rudy Mat. Inst. Steklov., 90 (1967), 209 pp. [2] R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452. [3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics 470, Springer, Berlin, 1975 [4] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848. [5] M. Brin, Ergodic theory of frame flows. Ergodic theory and dynamical systems, Ⅱ (College Park, MD, 1979/1980) (Progress in Mathematics, ) Birkhäuser, Boston, MA, 21 (1982), 163-183. [6] V. Climenhaga, S. Luzzatto and Y. Pesin, The geometric approach for constructing Sinai-Ruelle-Bowen measures, J. Stat. Phys., 166 (2017), 467-493.  doi: 10.1007/s10955-016-1608-7. [7] V. Climenhaga, Y. Pesin and A. Zelerowicz, Equilibrium states in dynamical systems via geometric measure theory, Bull. Amer. Math. Soc. (N.S.), 56 (2019), 569-610.  doi: 10.1090/bull/1659. [8] V. Climenhaga, Y. Pesin and A. Zelerowicz, Equilibrium measures for some partially hyperbolic systems, J. Mod. Dyn., 16 (2020), 155-205.  doi: 10.3934/jmd.2020006. [9] J. De Simoi and C. Liverani, Limit theorems for fast-slow partially hyperbolic systems, Invent. Math., 213 (2018), 811-1016.  doi: 10.1007/s00222-018-0798-9. [10] D. Dolgopyat, Lectures on $u$-Gibbs measures, http://www2.math.umd.edu/ dolgop/ugibbs.pdf. [11] N. Haydn and D. Ruelle, Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification, Commun. Math. Phys., 148 (1992), 155-167.  doi: 10.1007/BF02102369. [12] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 14 (1970), 133-163. [13] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, C.U.P., Cambridge, 1995. [14] B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math., 105 (1977), 81-105.  doi: 10.2307/1971026. [15] E. Mihailescu, Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets, Discrete Contin. Dyn. Syst., 32 (2012), 961-975.  doi: 10.3934/dcds.2012.32.961. [16] E. Mihailescu, Asymptotic distributions of preimages for endomorphisms, Ergodic Theory Dynam. Systems, 31 (2011), 911-934.  doi: 10.1017/S0143385710000155. [17] M. Misiurewicz, A short proof of the variational principle for a ZN+ action on a compact space, International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), Astérisque, No. 40, Soc. Math. France, Paris, (1976), 147–157. [18] S. Newhouse, Continuity properties of entropy, Ann. of Math., 129 (1989), 215-235.  doi: 10.2307/1971492. [19] S. Newhouse and T. Pignataro, On the estimation of topological entropy, J. Statist. Phys., 72 (1993), 1331-1351.  doi: 10.1007/BF01048189. [20] W. Parry, Bowen's equidistribution theory and the Dirichlet density theorem, Ergodic Theory Dynam. Systems, 4 (1984), 117-134.  doi: 10.1017/S0143385700002315. [21] W. Parry, Equilibrium states and weighted uniform distribution of closed orbits, Dynamical Systems, (College Park, MD, 1986–87), Lecture Notes in Math., Springer, Berlin, 1342 (1988), 617–625 doi: 10.1007/BFb0082850. [22] Y. Pesin and Y. Sinai, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.  doi: 10.1017/S014338570000170X. [23] D. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810. [24] Y. G. Sinai, Markov partitions and Y-diffeomorphisms, Funct. Anal. and Appl., 2 (1968), 64-89.  doi: 10.1007/BF01075361. [25] Y. G. Sinai, Gibbs measures in ergodic theory, Russian Math, Surveys, 27 (1972), 21-69.  doi: 10.1070/RM1972v027n04ABEH001383. [26] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1. [27] R. Spatzier and D. Visscher, Equilibrium measures for certain isometric extensions of Anosov systems, Ergodic Theory Dynam. Systems, 38 (2018), 1154-1167.  doi: 10.1017/etds.2016.62. [28] P. Walters, Ergodic Theory, Springer, 1982 [29] R. F. Williams, One-dimensional non-wandering sets, Topology, 6 (1967), 473-487.  doi: 10.1016/0040-9383(67)90005-5. [30] L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.

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##### References:
 [1] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Rudy Mat. Inst. Steklov., 90 (1967), 209 pp. [2] R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452. [3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics 470, Springer, Berlin, 1975 [4] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848. [5] M. Brin, Ergodic theory of frame flows. Ergodic theory and dynamical systems, Ⅱ (College Park, MD, 1979/1980) (Progress in Mathematics, ) Birkhäuser, Boston, MA, 21 (1982), 163-183. [6] V. Climenhaga, S. Luzzatto and Y. Pesin, The geometric approach for constructing Sinai-Ruelle-Bowen measures, J. Stat. Phys., 166 (2017), 467-493.  doi: 10.1007/s10955-016-1608-7. [7] V. Climenhaga, Y. Pesin and A. Zelerowicz, Equilibrium states in dynamical systems via geometric measure theory, Bull. Amer. Math. Soc. (N.S.), 56 (2019), 569-610.  doi: 10.1090/bull/1659. [8] V. Climenhaga, Y. Pesin and A. Zelerowicz, Equilibrium measures for some partially hyperbolic systems, J. Mod. Dyn., 16 (2020), 155-205.  doi: 10.3934/jmd.2020006. [9] J. De Simoi and C. Liverani, Limit theorems for fast-slow partially hyperbolic systems, Invent. Math., 213 (2018), 811-1016.  doi: 10.1007/s00222-018-0798-9. [10] D. Dolgopyat, Lectures on $u$-Gibbs measures, http://www2.math.umd.edu/ dolgop/ugibbs.pdf. [11] N. Haydn and D. Ruelle, Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification, Commun. Math. Phys., 148 (1992), 155-167.  doi: 10.1007/BF02102369. [12] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 14 (1970), 133-163. [13] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, C.U.P., Cambridge, 1995. [14] B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math., 105 (1977), 81-105.  doi: 10.2307/1971026. [15] E. Mihailescu, Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets, Discrete Contin. Dyn. Syst., 32 (2012), 961-975.  doi: 10.3934/dcds.2012.32.961. [16] E. Mihailescu, Asymptotic distributions of preimages for endomorphisms, Ergodic Theory Dynam. Systems, 31 (2011), 911-934.  doi: 10.1017/S0143385710000155. [17] M. Misiurewicz, A short proof of the variational principle for a ZN+ action on a compact space, International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), Astérisque, No. 40, Soc. Math. France, Paris, (1976), 147–157. [18] S. Newhouse, Continuity properties of entropy, Ann. of Math., 129 (1989), 215-235.  doi: 10.2307/1971492. [19] S. Newhouse and T. Pignataro, On the estimation of topological entropy, J. Statist. Phys., 72 (1993), 1331-1351.  doi: 10.1007/BF01048189. [20] W. Parry, Bowen's equidistribution theory and the Dirichlet density theorem, Ergodic Theory Dynam. Systems, 4 (1984), 117-134.  doi: 10.1017/S0143385700002315. [21] W. Parry, Equilibrium states and weighted uniform distribution of closed orbits, Dynamical Systems, (College Park, MD, 1986–87), Lecture Notes in Math., Springer, Berlin, 1342 (1988), 617–625 doi: 10.1007/BFb0082850. [22] Y. Pesin and Y. Sinai, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.  doi: 10.1017/S014338570000170X. [23] D. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810. [24] Y. G. Sinai, Markov partitions and Y-diffeomorphisms, Funct. Anal. and Appl., 2 (1968), 64-89.  doi: 10.1007/BF01075361. [25] Y. G. Sinai, Gibbs measures in ergodic theory, Russian Math, Surveys, 27 (1972), 21-69.  doi: 10.1070/RM1972v027n04ABEH001383. [26] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1. [27] R. Spatzier and D. Visscher, Equilibrium measures for certain isometric extensions of Anosov systems, Ergodic Theory Dynam. Systems, 38 (2018), 1154-1167.  doi: 10.1017/etds.2016.62. [28] P. Walters, Ergodic Theory, Springer, 1982 [29] R. F. Williams, One-dimensional non-wandering sets, Topology, 6 (1967), 473-487.  doi: 10.1016/0040-9383(67)90005-5. [30] L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.
The push forward of the measure $\lambda_n$ on $W_\delta^u(x)$ by $f^{k}$
A piece of local unstable manifold $W^u_\delta(x)$ of length $\ell$ for the Arnol'd CAT map and its image $f^nW^u_\delta(x)$
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
The push forward $f^m W_\delta^u(x)$ is $\epsilon$-dense
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