Abstract
We consider a class of \({\mathcal C}^{4}\) partially hyperbolic systems on \({\mathbb T}^2\) described by maps \(F_\varepsilon (x,\theta )=(f(x,\theta ),\theta +\varepsilon \omega (x,\theta ))\) where \(f(\cdot ,\theta )\) are expanding maps of the circle. For sufficiently small \(\varepsilon \) and \(\omega \) generic in an open set, we precisely classify the SRB measures for \(F_\varepsilon \) and their statistical properties, including exponential decay of correlation for Hölder observables with explicit and nearly optimal bounds on the decay rate.
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Notes
By a metastable system here we mean a situation in which two time scales are present: one, the short one, in which the system seems to have several invariant measures, and hence to lack ergodicity, and a longer time scale in which it turns out that the system has indeed only one relevant, mixing, invariant measure. This can be seen experimentally by the presence of two time scales in the decay of correlations.
That is, for \(\varrho _r\ne 0\), we let \(\omega \mapsto \varrho _r\omega \) and \(\varepsilon \mapsto \varrho _r^{-1}\varepsilon \) so that the product \(\varepsilon \omega \) is left unchanged, together with all other dynamically defined quantities. Observe that under this rescaling, (2.2) gives \(\psi _*\mapsto \varrho _r\psi _*\).
In [9] it is shown that assumption (A3) is in fact generic in \({\mathcal C}^2\). Observe moreover that the condition can be easily checked on periodic orbits.
In [30], \(\omega \) is said to be complete at \(\theta \) if this condition holds at \(\theta \).
The equivalence holds since the measures supported on periodic orbits are weakly dense in the set of the invariant measures [37].
The exact meaning of generic is a bit technical and we refer to [42] for the details.
We essentially prove that the Lyapunov exponents are negative, but this takes a good part of this paper.
The reader can easily work along the lines we suggest and construct more elaborate examples which are not skew-products, and yet feature all properties described in our examples.
It is possible to make this correspondence quantitatively precise for times of order \(\varepsilon ^{-\alpha }\) for some \(\alpha >0\). We refrain from doing it to keep the length of the paper under control and we postpone it to further work.
Of course, we mean this in the sense of [9, Theorem 2.8], on a scale smaller than \(\varepsilon \) the SRB could have some complicated fine structure. This issue is here left open.
Note that, by hypotheses, \(\rho _\varepsilon (1)=Z\varvec{\hat{\upsigma }}^{-2}e^{2\varepsilon ^{-1}\int _0^1\frac{{\bar{\omega }}}{\varvec{\hat{\upsigma }}^2}}=Z\varvec{\hat{\upsigma }}^{-2}=\rho _\varepsilon (0)\), hence \(\rho _\varepsilon \) is a smooth function on \({\mathbb T}\).
For the reader convenience, here is how to argue: compute using
$$\begin{aligned} 0\le \int _{{\mathbb T}}\left[ \varvec{\hat{\upsigma }}\left( e^{-V/2}\varphi \right) '\right] ^2 \text { and } \int _{{\mathbb T}} \varvec{\hat{\upsigma }}^2V'\varphi '\varphi e^{-V}=\frac{1}{2}\int _{{\mathbb T}}\left[ (\varvec{\hat{\upsigma }}V')^2-(\varvec{\hat{\upsigma }}^2V')'\right] \varphi ^2 e^{-V}. \end{aligned}$$Again, for the reader convenience, here is how to argue: first of all note that (3.5) implies that, on \(A_\varepsilon \) the ratio between the sup and inf to \(\rho _\varepsilon \) is bounded by \(e^{{c_\#}a^2}\), and remember that \(\nu _\varepsilon (A_\varepsilon )\ge 1/2\). Then
$$\begin{aligned} \int _{A_\varepsilon }\varphi ^2\rho _\varepsilon =\frac{1}{2\nu _\varepsilon (A_\varepsilon )}\int _{A_\varepsilon ^2}[\varphi (x)-\varphi (y)]^2\rho _\varepsilon (x)\rho _\varepsilon (y) dxdy+\frac{1}{\nu _\varepsilon (A_\varepsilon )}\left( \int _{A_\varepsilon }\varphi \rho _\varepsilon \right) ^2. \end{aligned}$$While \([\varphi (x)-\varphi (y)]^2\le (\int _{A_\varepsilon }|\varphi '|)^2\le |A_\varepsilon |\int _{A_\varepsilon }(\varphi ')^2\le {C_\#}\varepsilon \int _{{\mathbb T}}(\varphi ')^2\rho _\varepsilon \).
The choice of \(\epsilon \) is rather arbitrary, it suffices that it is small enough so that \({\mathcal T}_{\epsilon ,i}\ne \emptyset \). In the present case \(\epsilon =1/4\) will do.
Again, in the language of Sect. 6.4, one can take the neighborhood to be \(\bigcap _{\epsilon >0}{A}^+_{\epsilon ,\theta _{1,-}}\).
Of course, technically speaking, there is no attractor as condition (A4*) guarantees that the dynamics will visit an \(\varepsilon \)-dense set in configuration space. Yet, for small \(\varepsilon \) and each \(\beta \in (0,1/2)\) a portion \(1-e^{-{c_\#}\varepsilon ^{-1+2\beta }}\) of the mass is concentrated in a \({\mathcal O}(\varepsilon ^\beta )\)-neighborhood of \(\theta = 0\). So the situation differs indeed very little from an attractor. In passing, this example shows that a purely topological description of the dynamics can fail miserably in capturing the relevant properties of the motion.
This follows from a direct computation.
See also [9, Section 3.2] for a similar, but more general, account of the framework in this context.
Recall that a probability space is a Lebesgue space if it is isomorphic to the disjoint union of an interval [0, a] with Lebesgue measure and (at most) countably many atoms.
The set \(L_{c_1,c_2}\) of \((c_1,c_2)\)-standard pairs is in fact a space of smooth functions; it is thus a measurable space with the Borel \(\sigma \)-algebra. More in detail, if \({\mathbb G}:[a,b]\rightarrow {\mathbb T}^2\) and \(\rho :[a,b]\rightarrow {\mathbb R}^{+}\) are defined as above, let \(\hat{{\mathbb G}}\) and \(\hat{\rho }\) be defined by precomposing \({\mathbb G}\) and \(\rho \) respectively with the affine orientation-preserving map \([0,1]\rightarrow [a,b]\). A standard pair-valued function is thus \({\mathcal F}\)-measurable if both maps \((\alpha ,s)\mapsto \hat{{\mathbb G}}_{\alpha }(s)\) and \((\alpha ,s)\mapsto \hat{\rho }_{\alpha }(s)\) are jointly measurable. In particular, for any Borel set \(E\subset {\mathbb T}^2\), the function \(\alpha \mapsto \mu _{\ell (\alpha )}(E)\) is \({\mathcal F}\)-measurable.
This concept can be obviously applied to a single standard pair, considering it a family with just one element. In such case, the support of the standard pair and the support of the associated measure can be trivially identified.
The reader can easily fill in the details of the computations.
Obviously, \(\varvec{\pi }_{\mathcal {A}}(\alpha ,p)=\alpha \), for each \(\alpha \in \mathcal {A}, p\in {\mathbb T}^2\).
Notice that the definition of standard pair in fact depends on \(\varepsilon \), therefore this convergence holds for any sequence of standard pairs which in turn weakly converges to the flat standard pair \(\{\theta =\theta _*\}\).
The function \(\Delta z\) (and thus \(\Delta \theta \) and \(\Delta \zeta \)) indeed depend on \(\varepsilon \) (since so does \(z_\varepsilon \)); however, we do not explicitly add a subscript \(\varepsilon \) to ease notation.
Such lifts are uniquely determined by the condition \(\gamma _{A,\varepsilon }(0) = 0\) and \({\bar{\gamma }}_A(0) = 0\).
The function \({\bar{\omega }}^+(\theta )\) is continuous, therefore a solution of the given differential equation exists (by Cauchy–Peano Theorem), but in general is not unique.
Recall that \(H_k\) and \(\hat{H}_k\) are defined in Sect. 6.3.
In fact \(\hat{S}\) is a repelling set for the averaged dynamics, hence if \(\theta ^*_{\ell }\in {\mathbb S}{\setminus }S\), the averaged dynamics will certainly keep \(\theta \) away from \(\hat{S}\).
Observe that the choice of \(c_S\) depends on \(C_{T_{\mathrm{S}}}\) and thus on \(T_{\mathrm{S}}\).
Coupling has been long used in abstract Ergodic Theory under the name of joining, but it has been re-introduced in the study of the statistical properties of smooth systems (smooth Ergodic Theory) by Young [46], borrowing it from the theory of Markov chains. The version we are going to present here has been developed by Dmitry Dolgopyat in the standard pair framework.
We do not include explicitly \(\varvec{\mu }\) in the notation to make it more readable and as it does not create confusion.
Note that, with the above definition, \(\underline{\mathfrak {J}}^1\) will not be, in general, a standard family. Yet, it will be n-prestandard and that is all is needed in the following.
The reader can easily check that the differential equation admits a unique solution; recall (4.6) for the definition of \({s}_n\).
Recall that, according to Notational Remark 8.2, \(\mathcal {A}_{[k]}\) is the index set of \(\underline{{\mathfrak {L}}}_{[k]}\) and \({\upnu \;}_{[k]}\) the corresponding measure.
Remark that \(n_{Z,i}\) does not depend on \(\epsilon \), provided \(\epsilon \) has been chosen small enough.
Notice that by Lemma 6.13(d) \({\mathcal T}_{\epsilon ,i}\) contains a neighborhood of \(\theta _{i,-}\).
Such approximate functions can be obtained by standard mollification.
Note that there exists a natural measure-preserving immersion \(\mathbf {i}:\mathcal {A}_{[k+1]}\rightarrow \mathcal {A}_{[k]}\), thus one can always see \(U_{[k]}\) as a random variable on \(\mathcal {A}_{[k+1]}\) and similarly for the other random variables. It is thus possible to view all the relevant random variables on the same natural probability space (given by the last time at which we are interested). We will use this implicitly in the following.
The conditioning means simply that we specify the standard pair to which the process belongs at the iteration step \((j+1)p{\mathcal K}_\mathrm{A}{N_{\mathrm{S}}}\).
References
Alves, J.F., Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2), 351–398 (2000)
Alves, J.F., Luzzatto, S., Pinheiro, V.: Markov structures and decay of correlations for non-uniformly expanding dynamical systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(6), 817–839 (2005)
de Castro Junior, A.A.: Fast mixing for attractors with a mostly contracting central direction. Ergod. Theory Dyn. Syst. 24(1), 17–44 (2004)
Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Isr. J. Math. 115, 157–193 (2000)
Chernov, N., Markarian, R.: Chaotic billiards. In: Mathematical Surveys and Monographs, vol. 127. American Mathematical Society, Providence (2006)
Chernov, N.I.: Markov approximations and decay of correlations for Anosov flows. Ann. Math. (2) 147(2), 269–324 (1998)
Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth analysis and control theory. In: Graduate Texts in Mathematics, vol. 178. Springer, New York (1998)
de Castro Júnior, A.A.: Backward inducing and exponential decay of correlations for partially hyperbolic attractors. Isr. J. Math. 130, 29–75 (2002)
De Simoi, J., Liverani, C.: Limit theorems for fast-slow partially hyperbolic systems. arXiv:1408.5453 (Preprint)
Deuschel, J.-D., Stroock, D.W.: Large deviations. In: Pure and Applied Mathematics, vol. 137. Academic Press Inc, Boston (1989)
Dolgopyat, D.: On decay of correlations in Anosov flows. Ann. Math. (2) 147(2), 357–390 (1998)
Dolgopyat, D.: On dynamics of mostly contracting diffeomorphisms. Commun. Math. Phys. 213(1), 181–201 (2000)
Dolgopyat, D.: On mixing properties of compact group extensions of hyperbolic systems. Isr. J. Math. 130, 157–205 (2002)
Dolgopyat, D.: Limit theorems for partially hyperbolic systems. Trans. Am. Math. Soc. 356(4), 1637–1689 (2004) (electronic)
Dolgopyat, D.: On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155(2), 389–449 (2004)
Dolgopyat, D.: Averaging and invariant measures. Mosc. Math. J. 5(3), 537–576, 742 (2005)
Dolgopyat, D.: Bouncing balls in non-linear potentials. Discrete Contin. Dyn. Syst. 22(1–2), 165–182 (2008)
Dolgopyat, D.: Repulsion from resonances. Mémoires 128. Société mathématique de France, vi+119 (2012)
Dolgopyat, D., Liverani, C.: Energy transfer in a fast-slow Hamiltonian system. Commun. Math. Phys. 308(1), 201–225 (2011)
Field, M., Melbourne, I., Török, A.: Stability of mixing and rapid mixing for hyperbolic flows. Ann. Math. (2) 166(1), 269–291 (2007)
Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, 3rd edn. Springer, Heidelberg (2012). Translated from the 1979 Russian original by Joseph Szücs
Gouëzel, S.: Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. Fr. 134(1), 1–31 (2006)
Gouëzel, S., Liverani, C.: Banach spaces adapted to Anosov systems. Ergod. Theory Dyn. Syst. 26(1), 189–217 (2006)
Grayson, M., Pugh, C., Shub, M.: Stably ergodic diffeomorphisms. Ann. Math. (2) 140(2), 295–329 (1994)
Jenkinson, O.: Ergodic optimization. Discrete Contin. Dyn. Syst. 15(1), 197–224 (2006)
Jenkinson, O., Morris, I.D.: Lyapunov optimizing measures for \(C^1\) expanding maps of the circle. Ergod. Theory Dyn. Syst. 28(6), 1849–1860 (2008)
Kifer, Y.: Random perturbations of dynamical systems. In: Progress in Probability and Statistics, vol. 16. Birkhäuser Boston Inc., Boston (1988)
Kifer, Y.: Averaging in dynamical systems and large deviations. Invent. Math. 110(2), 337–370 (1992)
Kifer, Y.: Averaging principle for fully coupled dynamical systems and large deviations. Ergod. Theory Dyn. Syst. 24(3), 847–871 (2004)
Kifer, Y.: Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging. Mem. Am. Math. Soc. 201(944), viii+129 (2009)
Krylov, N.S.: Works on the Foundations of Statistical Physics. Princeton University Press, Princeton (1979). Translated from the Russian by A. B. Migdal, Ya. G. Sinai [Ja. G. Sinaĭ] and Yu. L. Zeeman [Ju. L. Zeeman], With a preface by A. S. Wightman, With a biography of Krylov by V. A. Fock [V. A. Fok], With an introductory article “The views of N. S. Krylov on the foundations of statistical physics” by Migdal and Fok, With a supplementary article “Development of Krylov’s ideas” by Sinaĭ, Princeton Series in Physics
Liverani, C.: On contact Anosov flows. Ann. Math. (2) 159(3), 1275–1312 (2004)
Liverani, C.: The 2009 Michael Brin Prize in dynamical systems. J. Mod. Dyn. 4(2), i–ii (2010)
Dolgopyat, D. Viana, M., Yang, J.: Geometric and measure-theoretical structures of maps with mostly contracting center. Comm. Math. Phys. 341(3), 991–1014 (2016). doi:10.1007/s00220-015-2554-y
Melbourne, I.: Rapid decay of correlations for nonuniformly hyperbolic flows. Trans. Am. Math. Soc. 359(5), 2421–2441 (2007) (electronic)
Olivieri, E., Vares, M.E.: Large deviations and metastability. In: Encyclopedia of Mathematics and its Applications, vol. 100. Cambridge University Press, Cambridge (2005)
Parthasarathy, K.R.: On the category of ergodic measures. Ill. J. Math. 5, 648–656 (1961)
Pesin, Y.: On the work of Dolgopyat on partial and nonuniform hyperbolicity. J. Mod. Dyn. 4(2), 227–241 (2010)
Pugh, C., Shub, M.: Stably ergodic dynamical systems and partial hyperbolicity. J. Complex. 13(1), 125–179 (1997)
Ruelle, D.: A mechanical model for Fourier’s law of heat conduction. Commun. Math. Phys. 311(3), 755–768 (2012)
Shub, M., Wilkinson, A.: Pathological foliations and removable zero exponents. Invent. Math. 139(3), 495–508 (2000)
Tsujii, M.: Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194(1), 37–132 (2005)
Tsujii, M.: Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity 23(7), 1495–1545 (2010)
Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202(950), iv+141 (2009)
Volk, D.: Private communication
Young, L.-S.: Recurrence times and rates of mixing. Isr. J. Math. 110, 153–188 (1999)
Young, L.-S.: What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108(5–6), 733–754 (2002). Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays
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This work would not exist without the many and fruitful discussions which both authors had with Dmitry Dolgopyat, who would very well deserve to be listed among the authors. We thank Ian Morris for suggesting the simple argument we use in the proof of Lemma 6.2. We also thank the anonymous referees for many very helpful suggestions among which to include the examples and discussion in Sect. 3, for pointing out several inaccuracies in previous versions of this paper and for suggesting a simpler and shorter argument for the proof of Theorem 6.4. The authors are also glad to thank Piermarco Cannarsa, Bastien Fernandez, Christophe Poquet and Ke Zhang for their very useful comments. This work was supported by the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953) and by NSERC. Both authors are pleased to thank the Fields Institute in Toronto, Canada, (where this work started) for the excellent hospitality and working conditions provided during the spring semester 2011.
Appendix: Random walks
Appendix: Random walks
We start by recalling a well known fact about one dimensional random walks (it can be obtained, e.g., from Cramer’s Theorem).
Lemma 11.1
Let \(\xi _k\in \{-1,1\}\) be a sequence of i.i.d. random variables with distribution \({\mathbb P}\left( \xi _i=1\right) =p\) for \(p\in (0,1)\). Let \(\mathbf {\Xi }_0=0\) and for \(n>0,\) define : \(\mathbf {\Xi }_n=\sum _{j=1}^n\xi _j\). For any \(c<2p-1\) there exist \(\vartheta ,\varrho \in (0,1)\) such that, for any \(k\in {\mathbb N}\) and \(a\in {\mathbb R}{:}\)
Next, we introduce an useful comparison argument:
Lemma 11.2
Let \(\xi _k\in \{-1,1\}\) be a sequence of independent random variables and let \(\eta _k\in \{-1,0,1\}\) be a random process such that
For \(n>0\) define the random variables
where \(N>0\) is some fixed natural number (if \(n=0\) we let them all equal to 0); then for each \(n\in {\mathbb N}\) and \(L\in {\mathbb Z}{:}\)
In particular, if \(\tau _\mathbf {\Xi }\) is the hitting time \(\tau =\inf \{k\,:\,\mathbf {\Xi }_k \ge L\}\) and \(\tau _\mathbf {H}=\inf \{k\,:\,\mathbf {H}_k \ge L\}\) we have, for any \(s>0{:}\)
Proof
(see [17, Proposition 2.4]) The proof amounts to design a suitable coupling \((\xi ^*_k,\eta ^*_k)\) of the random variables \(\xi _k\) and \(\eta _k\). Let us introduce an auxiliary sequence \(U_k\) of independent random variables uniformly distributed on [0, 1] and define the random variables
and
We then define
Clearly \(\xi ^*_k\) (resp. \(\eta ^*_k\)) has the same distribution of \(\xi _k\) (resp. \(\eta _k\)) and consequently \(\mathbf {\Xi }^*_k\) (resp. \(\mathbf {H}^*_k\)) has the same distribution of \(\mathbf {\Xi }_k\) (resp. \(\mathbf {H}_k\)). Moreover, \(\xi ^*_k\le \eta ^*_k\) by design which in turn implies that \(\mathbf {\Xi }^*_k\le \mathbf {H}^*_k\). This concludes the proof of our lemma.\(\square \)
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De Simoi, J., Liverani, C. Statistical properties of mostly contracting fast-slow partially hyperbolic systems. Invent. math. 206, 147–227 (2016). https://doi.org/10.1007/s00222-016-0651-y
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DOI: https://doi.org/10.1007/s00222-016-0651-y