On the Abundance of Sinai–Ruelle–Bowen Measures

Abstract

Motivated by Palis program (Ann Inst H Poincaré Anal Nonlinéaire 22:485–507, 2005) on physical measures, we prove that there is a dense set ${{\mathcal {D}}}$ in the space of $C^1$ diffeomorphisms such that for any diffeomorphism $f\in {{\mathcal {D}}}$, one of the following cases occurs: f admits at least one Sinai–Ruelle–Bowen measure; f has a physical measure supported on a sink; f has a homoclinic tangency. The key point is to prove the existence of a Sinai–Ruelle–Bowen measure for any $C^2$ partially hyperbolic attractor with one-dimensional dominated center bundles. To build such a measure, we consider well-controlled limits of smooth random perturbations. This gives Gibbs cu-states which generalize the Gibbs u-states of Pesin and Sinai. The partially hyperbolic splitting may contain many sub-bundles. So the Gibbs cu-states may associate to various possible center-unstable sub-bundles among the dominated splittings. Finally we analyze these Gibbs cu-states and their center-unstable indices to obtain the desired Sinai–Ruelle–Bowen measure.

Appendices

The Absolute Continuity of Invariant Manifolds

Let W be an embedded manifold of M. A foliation ${{\mathcal {F}}}$ of W is absolutely continuous if for any two cross section $\Sigma _1$ and $\Sigma _2$ in W that are close and transverse to the foliation ${{\mathcal {F}}}$ in W, the holonomy map $h:\Sigma _1\rightarrow \Sigma _2$ defined by the foliation ${{\mathcal {F}}}$ has the following property: $h_*(\mathrm{Leb}_{\Sigma _1})$ is absolutely continuous with respect to $\mathrm{Leb}_{\Sigma _2}$.

A fundamental property of an absolutely continuous foliation is the following (one can see [7, Lemma 3.4] for the proof):

Lemma A.1

Assume that W is an embedded sub-manifold of M and ${{\mathcal {F}}}$ is an absolutely continuous foliation of W. Then the conditional measures of the Lebesgue measure of W with respect to the measurable partition associated to ${{\mathcal {F}}}$ are absolutely continuous with respect to the Lebesgue measures of the leaves of ${{\mathcal {F}}}$.

About the plaque families, one has the following result (Lemma A.2) on the absolute continuity. Recall that

Lemma A.2

Assume that f is a $C^2$ diffeomorphism and assume that $\Lambda$ is a compact f-invariant set with a dominated splitting $TM|_\Lambda =\Delta _1\oplus _\succ \Delta _2\oplus _\succ \Delta _3$. Given $\ell \in {{\mathbb {N}}}$ and $\alpha >0$, there is $\delta =\delta (\ell ,\alpha )$ such that for any point $x\in \Lambda _\ell (\Delta _2,\alpha )$, i.e.,

$$\begin{aligned} \prod _{i=0}^{n-1}\Vert Df^{-\ell }|_{\Delta _2(f^{-i \ell }(x))}\Vert \le \mathrm{e}^{-\alpha \ell n},~\forall n\in {{\mathbb {N}}}, \end{aligned}$$

the foliation

$$\begin{aligned} \{W^{\Delta _1}_\delta (y):~y\in W^{\Delta _1\oplus \Delta _2}(x)\} \end{aligned}$$

is an absolutely continuous foliation of $W^{\Delta _1\oplus \Delta _2}(x)$.

Proof

We give a sketch of the proof. By relaxing the constants, for any point $x\in \Lambda _\ell (\Delta _2,\alpha )$, one has that

$$\begin{aligned} \prod _{i=0}^{n-1}\Vert Df^{-\ell }|_{\Delta _1\oplus _\succ \Delta _2(f^{-i \ell }(x))}\Vert \le \mathrm{e}^{-\alpha \ell n},~\forall n\in {{\mathbb {N}}}. \end{aligned}$$

Thus, by Lemma 4.4, there is $\delta =\delta (\ell ,\alpha )$ such that $W^{\Delta _1\oplus \Delta _2}_\delta (x)$ is contained in the (exponentially) unstable manifold of x.

Thus, by reducing $\delta$ if necessary, for any point $y\in W^{\Delta _1\oplus \Delta _2}_\delta (x)$, $W^{\Delta _1}_\delta (y)$ is the stronger unstable manifold in $W^{\Delta _1\oplus \Delta _2}_\delta (x)$. The absolute continuity follows from a similar argument in [8, Chapter 11]. $\square$

Now we can give the proof of Proposition 4.9.

Proof of Proposition 4.9

It suffices to prove that for any $0\le i\le k-1$, one has that ${{\mathcal {G}}}_i\supset {{\mathcal {G}}}_{i+1}$. We set $E=E^{uu}\oplus _\succ E_1^c\oplus _\succ \cdots \oplus _\succ E_{i+1}^c$ and $\Delta =E^{uu}\oplus _\succ E_1^c\oplus _\succ \cdots \oplus _\succ E_{i}^c$. For any measure $\mu \in {{\mathcal {G}}}_{i+1}$, one knows that:

• 1. $\mu$-almost every point has its Lyapunov exponents along E are positive.

• 2. The conditional measures of $\mu$ on $W^{E,u}$ are absolutely continuous w.r.t. Lebesgue.

By Item 1, there are $\ell \in {{\mathbb {N}}}$ and $\alpha >0$ such that $x\in \Lambda _\ell (\alpha ,E^c_{i+1})$. By Lemma A.2, the foliation

$$\begin{aligned} \{W^{\Delta }_\delta (y):~y\in W^{E}(x)\} \end{aligned}$$

is an absolutely continuous foliation of $W^{E}(x)$. Thus from Lemma A.1, the conditional measures of the Lebesgue measure on $W^E(x)$ on the foliation $\{W^{\Delta }_\delta (y):~y\in W^{E}(x)\}$ are Lebesgue measures. By Item 2 and the transitivity of conditional measures, one can conclude the proposition. $\square$

The Proof of the Pliss-Like Lemma

Proof of Lemma 6.8

For any given $\varepsilon >0$, take

$$\begin{aligned} 0<\rho < \min \left\{ 1,\frac{(\gamma _2-\gamma _1)}{2(2C-\gamma _1)}, \frac{\gamma _2-\gamma _1}{C-\gamma _1}\varepsilon \right\} . \end{aligned}$$

The subset ${{\mathbb {J}}}\subset {{\mathbb {N}}}$ is defined by

$$\begin{aligned} {{\mathbb {J}}}=\{j\in {{\mathbb {N}}}:~\sum _{i=0}^{n-1}a_{i+j}\le n\gamma _2,~~~\forall n\in {{\mathbb {N}}}\}. \end{aligned}$$

We are going to prove that $\limsup _{n\rightarrow \infty }\frac{1}{n}\#({{\mathbb {J}}}\cap [1,n])\ge 1-\varepsilon$. Fix $\gamma =(\gamma _1+\gamma _2)/2$.

Claim

For any L large enough, one has that

$$\begin{aligned} \sum _{i=1}^L a_i\le L\gamma . \end{aligned}$$

Proof

Choose a large integer $L\in {{\mathbb {L}}}$ such that $\rho L>1$. By the property of ${{\mathbb {L}}}$, there exists a set of integers ${{\mathbb {G}}}=\{n_1,n_2,\cdots ,n_k\}\subset [1,L]$ such that $\#{{\mathbb {G}}}\ge (1-\rho )L$ and $a_{n_i}<\gamma _1$. Thus, one has that

$$\begin{aligned} \sum _{i=1}^L a_i= & {} \sum _{m\in {{\mathbb {G}}}}a_m+\sum _{m\in [1,L]\setminus {{\mathbb {G}}}}a_m\le \gamma _1(1-\rho )L+C(\rho L+1) \\\le & {} ((2C-\gamma _1)\rho +\gamma _1)L\le \gamma L \end{aligned}$$

when $\rho <(\gamma _2-\gamma _1)/2(2C-\gamma _1)$. $\square$

From the above claim, by the usual Pliss Lemma as in [40], one knows that ${{\mathbb {J}}}$ is a non-empty set with infinite cardinality.

To conclude, it suffices to prove that for some large $J\in {{\mathbb {J}}}$, one has that ${{\mathbb {J}}}\cap [1,J]\ge (1-\varepsilon )J$. We will prove by contradiction and assume that ${{\mathbb {J}}}\cap [1,J]< (1-\varepsilon )J$ for any large J. Therefore, $[1,J]\setminus {{\mathbb {J}}}$ can be split into finitely many intervals $\{I_\alpha =[c_\alpha ,d_\alpha )\}_{\alpha \in {{\mathcal {A}}}}$ such that

• $\sum _{m\in [c_\alpha ,d_\alpha )}a_m\ge (d_\alpha -c_\alpha )\gamma _2$ for any $\alpha \in {{\mathcal {A}}}$.

• $\sum _{\alpha \in {{\mathcal {A}}}}(d_\alpha -c_\alpha )\ge \varepsilon J$.

Set ${{\mathbb {B}}}=\cup _{\alpha \in {{\mathcal {A}}}}I_\alpha$. Since $\liminf _{n\rightarrow +\infty }\frac{1}{n}\#\{[0,n-1]\cap {{\mathbb {L}}}\}>1-\rho$, for J large enough, one has that $\# ({{\mathbb {L}}}\cap [1,J])\ge (1-\rho )J$.

Claim

One has the following estimate:

$$\begin{aligned} \#({{\mathbb {B}}}\setminus {{\mathbb {L}}})\ge \frac{\gamma _2-\gamma _1}{C-\gamma _1}\#({{\mathbb {B}}})\ge \frac{\gamma _2-\gamma _1}{C-\gamma _1}\varepsilon J. \end{aligned}$$

Proof

We have the following two estimates:

• $\sum _{i\in {{\mathbb {B}}}}a_i>(\#{{\mathbb {B}}})\gamma _2$.

• $\sum _{i\in {{\mathbb {B}}}}a_i\le \sum _{i\in {{\mathbb {B}}}\cap {{\mathbb {L}}}}a_i+\sum _{i\in {{\mathbb {B}}}\setminus {{\mathbb {L}}}}a_i\le (\#({{\mathbb {B}}}\cap {{\mathbb {L}}}))\gamma _1+(\#({{\mathbb {B}}}\setminus {{\mathbb {L}}}))C=(\#{{\mathbb {B}}})\gamma _1+(\#({{\mathbb {B}}}\setminus {{\mathbb {L}}}))(C-\gamma _1).$

By combining the above two inequalities one obtains that $\#({{\mathbb {B}}}\setminus {{\mathbb {L}}})\ge \frac{\gamma _2-\gamma _1}{C-\gamma _1}\#({{\mathbb {B}}})$. The last inequality then follows from $\#{{\mathbb {B}}}\ge \varepsilon J$. $\square$

Consequently, we have that

$$\begin{aligned} \rho J\ge & {} \#([1,J]\setminus {{\mathbb {L}}})\ge \#({{\mathbb {B}}}\setminus {{\mathbb {L}}})\\\ge & {} \frac{\gamma _2-\gamma _1}{C-\gamma _1}\#({{\mathbb {B}}})\ge \frac{\gamma _2-\gamma _1}{C-\gamma _1}\varepsilon J. \end{aligned}$$

This gives a contradiction since $\rho < {(\gamma _2-\gamma _1)}\varepsilon /{(C-\gamma _1)}$. $\square$

Proof of the Addendum

Now we give the proof of the addendum in Sect. 1.

It has been announced by Araujo [5] that for a $C^1$ generic surface diffeomorphism f, either it has infinitely many sinks or it has finitely many hyperbolic attractors whose basins cover a full Lebesgue measure set of the manifold. Potrie [42] presented a complete proof of Araujo’s theorem by using the work of Pujals and Sambarino [43, Theorem B].

Recall that Qiu [44] has proved that for $C^1$ generic diffeomorphisms, any hyperbolic attractor supports a physical measure satisfying Pesin’s entropy formula. If f has only hyperbolic attractors, then one can apply Qiu’s result to conclude. If f has sinks, then the measures supported on sinks are physical measures that satisfy Pesin’s entropy formula.