1 Introduction↩︎

To study smooth ergodic theory of diffeomorphisms, the “homoclinic relation” has recently played an important role. The homoclinic relation between hyperbolic periodic orbits were first defined by Newhouse [1]. This notion has been extended to general hyperbolic ergodic measures; see, for instance, Buzzi-Crovisier-Sarig [2]. They proved that homoclinically related hyperbolic ergodic measures can be coded in one transitive countable Markov shift and established the local uniqueness of measures of maximal entropy. By controling the number of homoclinic classes via a dynamical Sard lemma, they confirmed Newhouse’s conjecture [3] on the finiteness of MMEs for \(C^{\infty}\) surface diffeomorphisms.

The sizes of stable and unstable manifolds provide direct information for establishing homoclinic relations, without using the dynamical Sard lemma. For a given hyperbolic ergodic measure, the existence of stable and unstable manifolds follows from Pesin’s theory [4]. However, the sizes of these manifolds depend on the measure, which does not suffice to establish the statistical properties for certain important hyperbolic ergodic measures. Buzzi-Crovisier-Sarig [5] proved that for measures that are close (qualitatively) to measures of maximal entropy, in the setting of \(C^\infty\) surface diffeomorphisms with positive topological entropy, the Pesin blocks can have uniformly large measure², and then they proved various statistical properties of measures of maximal entropy. This property is called strongly positive recurrence (SPR for short).

In this paper, we establish an effective SPR property for surface diffeomorphisms, along with several extended theorems. Specifically, we obtain in a quantitative way uniform sizes of stable and unstable manifolds simultaneously for all ergodic measures with large enough metric entropy. Let us point out that:

• We do not rely on the continuity of Lyapunov exponents. This allows us to prove the SPR property for \(C^\infty\) surface diffeomorphisms without relying on the main results of the preparatory work [6]. Moreover, we establish the continuity of Lyapunov exponents as a direct consequence of the SPR property.

• Our arguments give a perturbative version, i.e., for diffeomorphisms that are close the original one.

• It also applies to three-dimensional vector fields \(X: M\rightarrow TM\) without singularities. Our proof recovers Zang’s result [7] for measures of maximal entropy, and established the finiteness of the number of homoclinic classes for ergodic measures whose metric entropy is uniformly larger than \(h_{\rm top}(X)/2\), not just those of measures of maximal entropy. Moreover, we prove that the number of general ergodic equilibrium states is finite for admissible potentials \(\phi: M\rightarrow {\mathbb{R}}\) satisfying either \(\sup_{x\in M}|\phi(x)|< h_{\rm top}(X)/4\) or \(\sup\{\phi(x)-\phi(y)\;:\;x,y\in M\}< h_{\rm top}(X)/2\).

1.1 Long stable and unstable manifolds↩︎

Pesin blocks are sets on which stable and unstable manifolds are uniformly long. However, beyond Pesin theory, one may also obtain long stable and unstable manifolds at points which are typical for ergodic measures with large entropy. These observations enable us to establish homoclinic relations, for instance, Theorem 6.

Let \(M\) be a compact, boundaryless \(C^{\infty}\) Riemannian manifold of any dimension and let \(f: M\rightarrow M\) be a \(C^{r}, r>1\) diffeomorpism. By Oseledec’s theorem and Pesin’s theory, there is an \(f\)-invariant subset \(M'_{f}\) of full measure for all invariant measures, such that for every \(x\in M'_f\) \[T_xM=E^{s}(x,f) \oplus E^{c}(x,f) \oplus E^{u}(x,f),\] where \(E^{s/c/u}(x,f)\) corresponds to the subspace associated with negative/zero/positive Lyapunov exponents respectively. When the diffeomorphism \(f\) is fixed, we denote these spaces simply by \(E^{s/c/u}(x)\). For each \(x\in M'_f\), the unstable manifold \[\label{eq:32dofWu} W^u(x):=W^u(x,f)=\big\{y\in M: \limsup_{n\to\infty} \frac{1}{n} \log d( f^{-n}(y), f^{-n}(x) )<0 \big\}.\tag{1}\] and the stable manifold \[\label{eq:32dofWs} W^s(x):=W^s(x,f)=\big\{y\in M: \limsup_{n\to\infty} \frac{1}{n} \log d( f^{n}(y), f^{n}(x) )<0 \big\}\tag{2}\] are \(C^r\) immersed manifolds. For \(\beta>0\), let \(E^{u/s}(x)(\beta):=\{v\in E^{u/s}(x): \| v \|\leq \beta \}\). We consider the sets of points with unstable/stable manifolds of size \(\beta\).

Note that the whole manifold \(M\) may have arbitrary dimension. When the unstable manifold is one-dimensional, we have the following result:

Theorem 1 provides more detailed informations than the main results in [8]. In [8], even when the metric entropy of an ergodic measure is close to the topological entropy, one only knows that \(\mu(L^u_{\beta_0}(f))\) is bounded away from zero; one does not know whether it can be close to \(1\). However, to establish the homoclinic relation, it is preferable that \(\alpha_2\) is close to \(1\); at the very least, \(\alpha_2\) should be greater than \(1/2\). This fact will be applied to prove the finiteness of the number of homoclinic classes (the equivalence classes under homoclinic relation of all hyperbolic ergodic measures, see Section 2.3).

To get the uniform size of stable and unstable manifolds simultanously, by adding some additional information in Theorem 1, one has the following Theorem 2.

We expect that a similar statement should also hold under the weaker assumption \(\alpha_2>0\), rather than only in the case \(\alpha_2>\tfrac{1}{2}\).

The information on unstable manifolds (nonlinear conditions) also yields information about their derivatives (linear conditions). For \(1 > \alpha_1 > \alpha_2 > 0\), define \[\label{eq:chi12} \chi(\alpha_1,\alpha_2):=h_{\rm top}(f) \cdot \frac{\alpha_1-\alpha_2}{1-\alpha_2}.\tag{3}\]

It is clear that Theorem 1 follows from Theorem 3. When \(\alpha_2\) is close to one and the stable manifold is also one-dimensional, a stronger result can be obtained; see Theorem 4. These results can be applied to establish SPR properties for surface diffeomorphisms, or three-dimensional flows. We first introduce the following sets, which we call quasi-hyperbolic sets.

Clearly, Theorem 2 is a direct consequence of Theorem 4. Therefore, to complete the proofs of all the results in this subsection, it suffices to prove Theorems 3 and 4.

1.2 Surface diffeomorphisms↩︎

The main theorems above can be applied to the surface case. Theorem 5 gives a quantitative and perturbative version of strongly positively recurrent property for surface diffeomorphisms. The notation \({\rm PES}_{\ell}^{\chi,\varepsilon}\) will be introduced in Section 2.1.

We recall below the definition of the SPR property introduced by Buzzi-Crovisier-Sarig [5].

In fact, the SPR property defined in [5] does not require the limit to approach \(1\); it only requires the liminf to be positive.

As a direct corollary of Theorem 5, we prove that for every \(C^{\infty}\) surface diffeomorphism \(f\) with \(h_{\rm top}(f)>0\), there exist \(\chi>0\) and \(C_{\chi}>0\) such that for every \(\varepsilon>0\) and every \(0<\tau<\frac{1}{4}\) \[\lim_{\ell \rightarrow \infty} ~\inf \big\{\mu({\rm PES}_{\ell}^{\chi,\varepsilon}(f)):~\mu~\text{is ergodic and}~h_{\mu}(f)>(1-\tau)h_{\rm top}(f)\big\} \geq 1-\frac{C_\chi}{\varepsilon} \cdot \tau.\] In particular, letting \(\tau \rightarrow 0\), we get that every \(C^{\infty}\) surface diffeomorphism with positive topological entropy is SPR. This provides another proof of [5], as our proof does not rely on the continuity of Lyapunov exponents in the preparatory work [6]. The key ingredient in our proof is a new upper bound for the metric entropy, expressed as a linear combination involving the topological entropy and the Lyapunov exponents, where the coefficient of the topological entropy is the measure of a set with good hyperbolicity, see 18 . To show this upper bound we use a variant of Yomdin’s theory [9], [10] introduced in [11].

1.3 Three dimensional vector fields↩︎

The suspension of a \(d\)-dimensional diffeomorphism yields a \((d+1)\)-dimensional flow. However, flows may not admit a global cross section, so understanding their dynamics requires additional methods beyond those for diffeomorphisms.

Ergodic theory of surface diffeomorphisms has been extensively studied by Sarig [12] and Buzzi-Crovisier-Sarig [2], [5]. There are some progresses for 3-D flows, Gan-Yang [13], Lima-Sarig [14], Buzzi-Crovisier-Lima [15] and Zang [7]. However, in contrast to surface diffeomorphisms, the corresponding theory for three-dimensional flows is not yet well established. The main theorems presented above provide new progress in this direction. For a flow \((\varphi^t)_{t\in\mathbb{R}}\) generated by a vector field \(X:M\rightarrow TM\) over a three-dimensional compact Riemannian manifold \(M\) without boundary, one can also define the homoclinic equivalent class \({\rm HEC}(\nu)\) of an hyperbolic measure \(\nu\) (see Section 2.3) and Pesin uniform sets (see Section 2.1). We first prove the finiteness of homoclinic equivalent classes with large entropy in the following setting.

Recall that, for every ergodic hyperbolic measure \(\mu\) of \((\varphi^t)_{t\in\mathbb{R}}\), there is at most one ergodic hyperbolic measure \(\nu\) which is homoclinically related to \(\mu\) that maximizes the entropy, or, more generally, maximizes the topological pressure \(P_{X}(\cdot)\) for admissible potentials such as Hölder continuous functions or geometric potentials [15]. Thus, we can directly obtain the following corollary, since the role of ?? is to guarantee that every equilibrium measure has entropy uniformly larger than \(h_{\rm top}(X)/2\).

Note that measures of maximal entropy are the equilibrium measures for the potential \(\phi \equiv 0\). Thus, Corollary 7 implies the main result of Zang [7].The final theorem provides an estimate for the measure of Pesin set for three-dimensional vector fields without singularities, thereby establishing an effective SPR property for such vector fields.

2 Preliminaries↩︎

2.1 Pesin sets↩︎

We first recall the definition of Pesin sets³. Let \(M\) be a compact \(C^{\infty}\) Riemannian manifold of any dimension, and let \(f: M\rightarrow M\) be a \(C^{1}\) diffeomorphism.

For every ergodic measure \(\mu\) of \(f\), if the non-zero Lyapunov exponents of \(\mu\) lie outside the interval \([-\chi,\chi]\), then we have \(\mu( {\rm NUP}_{\chi}(f) )=1\). We denote by \({\rm m}(A):=\min\{\|Av\|:\|v\|=1\}\) the co-norm of the linear operator \(A\).

By classical Pesin’s non-uniformly hyperbolic theory, for every ergodic hyperbolic measure \(\mu\) of \(f\) with \(\mu({\rm NUP}_{\chi}(f))=1\) and every \(0<\varepsilon\ll \chi\), one has \(\mu\left(\bigcup_{\ell>0} {\rm PESC}_{\ell}^{\chi,\varepsilon}(f)\right)=1\).

Recall Definition 2 of the quasi-hyperbolic set. In the non-uniformly hyperbolic setting, the measure of a Pesin set is closely related to the measure of the corresponding quasi-hyperbolic set. Buzzi-Crovisier-Sarig showed the following result in [5].

Pesin sets can also be defined in the setting of vector fields. Assume that \(X:M\rightarrow TM\) is a \(C^1\) vector field without singularities and \((\varphi^t)_{t\in {\mathbb{R}}}\) is the flow generated by \(X\). Note that the Lyapunov exponent in the flow direction is zero. For \(0<\varepsilon \ll \chi\) and \(\ell \geq 1\), we denote \[{\rm NUP}_{\chi}(X)={\rm NUP}_{\chi}(\varphi^1), ~~{\rm PES}_{\ell}^{\chi,\varepsilon}(X):={\rm PESC}_{\ell}^{\chi,\varepsilon}(\varphi^1).\]

The difference between Lemma 10 and [5] is the additional consideration of the center direction and the angle between subspaces. The estimate for the center direction can be obtained by using a similarly argument as in [5]. The estimate on the angle between subspaces follows from the fact that \[\lim_{n\rightarrow \pm \infty} \frac{1}{n} \log \angle \big(E^{\tau_1}(f^n(x)), E^{\tau_2}(f^n(x))\big)=0,~\tau_1,\tau_2 \in \{u,c,s\},~\tau_1 \neq \tau_2.\] We omit the proofs of of Lemma 8 and Lemma 10, which can be found in [5] and [7], respectively.

2.2 Metric entropy and topological entropy↩︎

Let \(f\) be a diffeomorphism on a compact Riemannian manifold \(M\). Denote by \(d\) the Riemannian distance on \(M\). For \(x\in M\), \(n\in \mathbb{N}\) and \(\varepsilon>0\), define the \((n,\varepsilon,f)\)-Bowen ball at \(x\) by \[B_n(x,\varepsilon,f):=\{y\in M:~d(f^i(y),f^i(x))<\varepsilon,~\forall 0\leq i<n\}.\] We sometimes use the simplified notations \(B_n(x,\varepsilon)=B_n(x,\varepsilon,f)\) when there is no confusion. A subset \(Y\subset M\) is said an \((n,\varepsilon)\)-spanning set, if \(M=\bigcup_{z\in Y}B_n(z,\varepsilon).\) We denote by \(r(f,n,\varepsilon)\) the minimal cardinality of all possible \((n,\varepsilon)\)-spanning sets. The topological entropy is thus defined to be \[h_{\rm top}(f):=\lim_{\varepsilon\to 0}\limsup_{n\to\infty}\frac{1}{n}\log r(f,n,\varepsilon).\] For a probability measure \(\mu\) (not necessarily invariant) and a finite partition \(\mathcal{P}\), define the static entropy of \(\mu\): \[H_\mu(\mathcal{P})=\sum_{P\in\mathcal{P}}-\mu(P)\log\mu(P)=\int -\log\mu(P(x)){\rm d}\mu(x).\] For the diffeomorphism \(f\), one defines \({\mathcal{P}}^n:={\mathcal{P}}^{n,f}=\bigvee_{j=0}^{n-1}f^{-j}(\mathcal{P})\). For an invariant measure \(\mu\), the metric entropy of \(\mu\) with respect to a partition \(\mathcal{P}\) is \[h_\mu(f,\mathcal{P})=\lim_{n\to\infty}\frac{1}{n}H_\mu({\mathcal{P}}^n);\] and the metric entropy of \(\mu\) is defined to be \[h_\mu(f)=\sup \big\{h_\mu(f,\mathcal{P}): {\cal P}~\text{is a finite partition} \big\}.\] We introduce the following proposition, which follows from [8] or Lemma 2 in [16]. Let \(\xi\) be a measurable partition subordinate to \(W^u\) with respect to \(f\) (we refer to [17] for the definition and the standard properties of subordinate partitions).

Assume that \(X:M\rightarrow TM\) is a \(C^1\) vector field and \((\varphi^t)_{t\in {\mathbb{R}}}\) is the flow generated by \(X\). For an invariant measure \(\mu\) of the flow \((\varphi^t)_{t\in {\mathbb{R}}}\), we define the metric entropy of \(\mu\) of \(X\) by \(h_{\mu}(X)=h_{\mu}(\varphi^1)\). Let \(\mu\) be an ergodic measure \(\mu\) of the \((\varphi^t)_{t\in {\mathbb{R}}}\). Note that \(\mu\) may not be an ergodic measure of \(\varphi^1\). But every ergodic component \(\nu\) of \(\mu\) with respect to \(\varphi^1\) satisfies \(h_{\nu}(\varphi^1)=h_{\mu}(X)\).

2.3 Homoclinic equivalent classes of ergodic hyperbolic measures↩︎

2.3.1 Surface diffeomorphisms↩︎

We first recall the definition of the homoclinic relation for ergodic hyperbolic measures of surface diffeomorphisms (see [2]). Let \(f\) be a \(C^r,r>1\) surface diffeomorphism. For two ergodic hyperbolic measures \(\mu_1\) and \(\mu_2\), we say that \(\mu_1\) is homoclinicaly related with \(\mu_2\), if there exist measurable sets \(\Lambda_1,\Lambda_2\) with \(\mu_1(\Lambda_1)>0\) and \(\mu_2(\Lambda_2)>0\)⁴ such that for every \(x\in \Lambda_1\) and every \(y\in \Lambda_2\), there exist \(n_1,n_2,k_1,k_2 \in {\mathbb{Z}}\) for which \[W^u(f^{n_1}(x)) \pitchfork W^s(f^{n_2}(y))\neq \emptyset,~W^s(f^{k_1}(x)) \pitchfork W^u(f^{k_2}(y))\neq \emptyset.\] Let \(\mu\) be a hyperbolic ergodic measure, the homoclinic equivalent classes of \(\mu\) is defined by \[{\rm HEC}(\mu):= \big\{\nu:~\nu~\text{is an ergodic hyperbolic measure that is homoclinically related with}~\mu \big\}.\] Here the homoclinic equivalence classes form a subset of all ergodic hyperbolic measures. Recently, Buzzi-Crovisier-Sarig [2] proved for any \(a>0\) the finiteness of the number of homoclinic classes for ergodic measures of a \(C^{\infty}\) surface diffeomorphism with entropy larger than \(a\):

Recall the Definition 1 of long unstable manifolds. Long stable and unstable manifolds make it easier to build homoclinic relations. The following proposition follows directly from the definition of homoclinic classes and the compactness of the whole manifold. Since its proof is analogous to that of Proposition 14, we omit the details.

2.3.2 Three-dimensional vector fields↩︎

Suppose that \(M\) is a three-dimensional compact Remiannian manifold without boundary. Let \(X:M \rightarrow TM\) be a \(C^{r},r>1\) vector field without singularities and \((\varphi^t)_{t\in {\mathbb{R}}}\) be the flow generated by \(X\). One can define a notion of homoclinic relation between ergodic hyperbolic measures (see also [15]). For every regular point \(x\in M\), we define the weak-stable and weak-unstable manifold at \(x\) by \[W^s({\rm Orb}(x))=\bigcup_{t\in \mathbb{R}} W^s(\varphi^t(x)),~W^u({\rm Orb}(x))=\bigcup_{t\in \mathbb{R}} W^u(\varphi^t(x))\] where \(W^{u/s}(x):=W^{u/s}(x,\varphi^1)\) are as defined in 1 and 2 , and are one-dimensional immersed sub-manifolds, respectively. Then, for two ergodic hyperbolic measures \(\mu_1\) and \(\mu_2\) of the flow \((\varphi^t)_{t\in {\mathbb{R}}}\), we say that \(\mu_1\) is homoclinically related with \(\mu_2\), if there exist measurable sets \(\Lambda_1, \Lambda_2\) with \(\mu_1(\Lambda_1)>0\) and \(\mu_2(\Lambda_2)>0\) such that \[W^u({\rm Orb}(x)) \pitchfork W^s({\rm Orb}(y)) \neq \emptyset,~~W^u({\rm Orb}(y)) \pitchfork W^s({\rm Orb}(x)) \neq \emptyset,~~\forall x\in \Lambda_1,~\forall y\in \Lambda_2.\] Let \(\mu\) be an ergodic hyperbolic measure of the flow \((\varphi^t)_{t\in {\mathbb{R}}}\). By [15], the homoclinic relation is an equivalence relation among ergodic hyperbolic measures of \((\varphi^t)_{t\in {\mathbb{R}}}\). The homoclinic equivalent classes of \(\mu\) is then defined as \[{\rm HEC}(\mu):=\big\{\nu:~\nu~\text{is an ergodic hyperbolic measure that is homoclinically related with}~\mu \big\}.\]

For \(x,y\in M\) with \(d(x,y)\) less than the radius of injectivity \(r(M)\) of \(M\) and for subspaces \(E\subset T_xM\) and \(F\subset T_yM\), we let \(\angle (E,F):=\min \{\angle (E, D_y(\exp_x^{-1})(F)), \angle (F, D_x(\exp_y^{-1})(E))\}\). We also define the Lipschitz constant of \(X\) as the least positive real number \(L>0\) satisfying \(\|d_y(\exp_x^{-1}) X(y)- X(x)\|, \|d_x(\exp_y^{-1})X(x)- X(y)\|\leq Ld(x,y)\) for any \(x,y\) with \(d(x,y)<r(M)\).

2.4 Raparametrization Lemmas↩︎

Let \(M\) be a compact Riemannian manifold without boundary of any dimension. Recall that a \(C^r\) curve \(\sigma:[-1,1]\rightarrow M\) is called bounded if \[\sup_{2\leq s\leq r} \|D^s \sigma\|_{\sup} \leq \frac{1}{6} \|D\sigma\|_{\sup}.\] If \(\sigma\) is a bounded curve, then \(\|D\sigma\|_{\sup}\leq 2\|D_t\sigma \|\) and \(\sin \angle(D_{s}\sigma, D_{t}\sigma) <\frac{1}{2}\) for any \(t,s\in [-1,1]\). A bounded curve is said to be strongly \(\varepsilon\)-bounded, if \(\|D\sigma\|_{\rm sup}\le\varepsilon\). A map \(\theta:[-1,1]\rightarrow [-1,1]\) is called a affine reparametrization if \(\theta'\) is constant and positive. Denote \(\sigma_*:=\sigma([-1,1])\).

We have the following local reparametrization from [16].

Let \(\sigma:[-1,1]\rightarrow M\) be a bounded curve. Let \(g:M\rightarrow M\) be a \(C^r\) diffeomorphism, we define the set \[\Sigma(\{\chi^{+}_k, \chi_k\}_{k=1}^{n})\subset \sigma_*\] as follows: \(z \in \Sigma(\{\chi^{+}_k, \chi_k\}_{k=1}^{n})\) if and only if for every \(1\le k \le n\) \[\lceil \log\|D_{g^{k-1}(z)}g\| \rceil=\chi^+_k,~~\lceil \log\|D_{g^{k-1}(z)}g|_{T_{g^{k-1}(z)}{(g^{k-1}\circ \sigma)_*}}\| \rceil=\chi_k.\] Meanwhile, for any \(z\in \sigma_*\), one defines \[\chi^+_{k}(z):=\lceil \log\|D_{g^{k-1}(z)}g\| \rceil,~~\chi_{k}(z):=\lceil \log\|D_{g^{k-1}(z)}g|_{T_{g^{k-1}(z)}{(g^{k-1}\circ \sigma_*)}}\| \rceil.\]

3 Entropy and the measure of Pesin sets↩︎

In this section, we provide the proofs of Theorems 3, 4 and 5. We begin with a key proposition that serves as the main ingredient in the proofs of these theorems. We first introduce some notations.

Assume that \(f:M\rightarrow M\) is a \(C^r,~r>1\) diffeomorphism. Given \(1>\alpha_1>\alpha_2>0\) and \(0<\chi_0<\chi<\chi(\alpha_1,\alpha_2)\). Recall the choice of \(\chi(\alpha_1,\alpha_2)\) in 3 , we have \[\frac{\alpha_1 h_{\rm top}(f)-\chi}{h_{\rm top}(f)-\chi}>\frac{\alpha_1 h_{\rm top}(f)-\chi(\alpha_1,\alpha_2)}{h_{\rm top}(f)-\chi(\alpha_1,\alpha_2)}=\alpha_2.\] Therefore, we can choose a small number \(\alpha>0\) such that \[\label{eq:choiceofalpha} \frac{\alpha_1 h_{\rm top}(f)-\chi-2\alpha}{h_{\rm top}(f)-\chi+\alpha}>\alpha_2.\tag{6}\]

Choose a large integer \(q\in {\mathbb{N}}\) such that (recall the number \(C_r>1\) in Lemma 16) \[\label{eq:32Q} \frac{2}{q}\log (5qC_r A_f)+\frac{1}{q(r-1)}<\alpha,~~\text{where}~A_f:=2\log(1+\|Df\|_{\sup}+\|Df^{-1}\|_{\sup}).\tag{7}\] Choose a constant \(\Upsilon:=\Upsilon_{q}>0\) and a \(C^r\) neighborhood \(\mathcal{U}_{1}\) of \(f\) such that for any \(g\in \mathcal{U}_1\), one has \[\label{eq:ChooseUp} \max_{1\leq j \leq r} \{\|D^j g^{k}\|_{\sup},~\|D^j g^{-k}\|_{\sup} \}<\Upsilon,~\forall 1\leq k \leq q;~~4A_g \leq 5A_f,~~ \|Dg^{\pm}\|_{\sup}\leq 2 \|Df^{\pm}\|_{\sup}.\tag{8}\] Choose \(\varepsilon(\Upsilon)>0\) as in Lemma 16. Take a positive number \(\varepsilon<\varepsilon(\Upsilon)/10\) and an integer \(N \in {\mathbb{N}}\), such that \[\label{eq:U} 10\cdot \varepsilon \cdot \|Df\|_{\sup}^q<r(M) \text{ and}~~\forall n\geq N, \;\log r\big(f^q,n,\frac{\varepsilon}{4} \big) < q\cdot n \left(h_{\rm top}(f)+\frac{\alpha}{2}\right),\tag{9}\] where \(r(f^q,n,\frac{\varepsilon}{4})\) denotes the minimal number of \((n,\frac{\varepsilon}{4},f^q)\)-Bowen balls which can cover \(M\), and \(r(M)\) is a positive constant such that \(\exp_x\) maps \(T_xM(2r(M))\) diffeomorphically onto a neighborhood of \(B(x,r(M))\) for every \(x\in M\). There is a \(C^r\) neighborhood \(\mathcal{U}\subset \mathcal{U}_{1}\) of \(f\) such that for every \(g\in \mathcal{U}\) \[\label{eq:U1} 10\cdot \varepsilon \cdot \| Dg \|_{\mathrm{sup}}^q<r(M) \; \text{and} \; \forall n=N, \cdots, 2N, \;\log r\big(g^q,n,\frac{\varepsilon}{4}\big) <\log r\big(f^q,n,\frac{\varepsilon}{4}\big)+nq \cdot \frac{\alpha}{2}.\tag{10}\] Then, for any \(n\geq N\) we get with \(R=n-N(\lceil n/N\rceil -1)\in [N,2N]\) \[\label{eq:N} \begin{align} \log r\big(g^q,n,\frac{\varepsilon}{2}\big)&\leq (\lceil n/N \rceil -1)\log r\big(g^q,N,\frac{\varepsilon}{4}\big)+\log r\big(g^q,R,\frac{\varepsilon}{4}\big) \\ &\leq q\cdot n (h_{\rm top}(f)+\alpha). \end{align}\tag{11}\]

Recall that \(\chi<\chi(\alpha_1,\alpha_2)<\alpha_1 h_{\rm top}(f)\). For \(g\in \mathcal{U}\), we consider the sets \[\begin{align} &H_{\chi}^u(g^{q}):= \{x : \| D_{x}g^{-qn}|_{E^{u}(x)} \|\leq e^{-qn\chi},~\forall n\geq 0 \}; \tag{12} \\ &H_{\chi}^s(g^{q}):= \{x : \| D_{x}g^{qn}|_{E^{s}(x)} \|\leq e^{-qn\chi},~\forall n\geq 0 \}. \tag{13} \end{align}\] For each ergodic measure \(\nu\) of \(g^q\) with \(h_{\nu}(g^q)\geq q\alpha_1 h_{\rm top}(f)>q\chi\), one has (see [16]) \[\nu(H_{\chi}^u(g^{q}))>0,~~\nu(H_{\chi}^{s}(g^{q}))>0.\] Choose \(\beta:=\beta_{\varepsilon}\) satisfies the conclusion of Lemma 15. Recall the Definition 1 of \(L^u_{\beta}(g)\) and \(L^s_{\beta}(g)\). We denote \[\begin{align} &\Lambda^{u,\chi}_{N,\beta}(g^q)=\{x: \exists \, 0\leq i\leq N~\text{s.t.}~g^{qi}(x)\in L^u_{\beta}(g)\cap H_{\chi}^{u}(g^q)\}; \tag{14} \\ &\Lambda^{s,\chi}_{N,\beta}(g^q)=\{x: \exists \, 0\leq i\leq N~\text{s.t.}~g^{-qi}(x)\in L^s_{\beta}(g)\cap H_{\chi}^{s}(g^q)\}. \tag{15} \end{align}\] We are going to prove that (which will be deduced from 18 ) \[\mu(\Lambda^{u,\chi}_{N,\beta}(g^q))\gtrapprox \frac{h_{\mu}(g)-\chi}{h_{\rm top}(f) -\chi}.\] The main proposition of this section is stated as follows.

We first provide the proof of Theorem 3, Theorem 4 and Theorem 5 by assuming that Proposition 18 holds, and then give the proof of Proposition 18.

3.1 Proof of Theorem 3 and Theorem 4↩︎

The proof of Theorem 3 essentially follows from Proposition 18 and the discussion at the beginning of this Section. Below, we provide the complete proof.

Next, we prove Theorem 4. The main issue here is the angle between subspaces. We first prove the following lemma.

3.2 Measure estimates for Pesin Sets in surface diffeomorphisms↩︎

Now, we prove Theorem 5, which can be almost directly deduced from Theorem 4.

3.3 Proof of Proposition 18↩︎

We give a lower bound for the measure of \(\Lambda^{u,\chi}_{N,\beta}(g^q)\); the lower bound for the measure of \(\Lambda^{s,\chi}_{N,\beta}(g^q)\) can be obtained by considering \(g^{-1}\). For convenience, we denote \(G:=g^q\) and \[\label{eq:Lambdas} \Lambda:=\Lambda^{u,\chi}_{N,\beta}(g^q)=\{x: \exists \, 0\leq i\leq N~\text{s.t.}~g^{qi}(x)\in L^u_{\beta}(g)\cap H_{\chi}^{u}(G)\}.\tag{17}\] Since \(\mu\) may not to be ergodic of \(G\), we consider the ergodic component \(\nu\) of \(\mu\) with respect to \(G\). Note that \(\mu=\frac{1}{q}\sum_{j=0}^{q-1}g_*^j\nu\). We want to show that \[\label{eq:key} h_{\mu}(g)=\big(h_{\nu}(g^q)/q\big)\lessapprox \nu(\Lambda)h_{\rm top}(f)+(1-\nu(\Lambda))\chi.\tag{18}\] in the following Proposition.

3.4 Proof of Proposition 21↩︎

3.4.1 Choice of the uniform set \(K\) and a strongly \(\varepsilon\)-bounded curve \(\sigma\)↩︎

Recall that \(G=g^q\) and \(\nu\) is an ergodic measure of \(G\). Let \(K\) be as in Proposition 11. We may find a compact subset \(K'\) of \(K\) satisfying the following properties:

1. \(\nu(K')>0\) and \(K' \subset H_{\chi}^u(G)\), where \(H_{\chi}^u(G)\) is defined in 12 ;

2. \(\frac{1}{n} \# \{0\leq i < n: G^{i}(x)\in \Lambda\}\rightarrow \nu( \Lambda)\) uniformly in \(x\in K'\);

3. \(\frac{1}{n}\log \|D_xG^{n}|_{E^u (x)}\|\rightarrow \lambda^{+}(\nu,G)\) uniformly in \(x\in K'\);

4. \(\frac{1}{n} \sum_{i=0}^{n-1} \delta_{G^{i}(x)} \rightarrow \nu\) uniformly in \(x\in K'\).

Fix a point \(x_0\in K'\) and choose a strongly \(\varepsilon\)-bounded curve \(\sigma:[-1,1]\rightarrow W^u(x_0)\) with \(\sigma(0)=x_0\) and \(\nu_{\xi(x_0)}( \sigma_{\ast}\cap K')>0\). We consider a sequence of finite partitions \(\{{\cal P}_k\}_{k\in {\mathbb{N}}}\), satisfying \(\nu(\partial {\cal P}_k)=0\) and \({\rm Diam}({\cal P}_k)\xrightarrow{k\to \infty} 0\). Then, by Proposition 11 we have with \({\cal P}_k^n:={\cal P}_k^{n,G}\) \[\label{e46entropy-cardinality} h_{\nu}(G)\leq \liminf_{k\rightarrow \infty} \left(\liminf_{n\rightarrow \infty}\frac{1}{n} \log \# \{ P\in {\cal P}_k^n : P\cap K' \cap \sigma_*\neq \emptyset\} \right).\tag{20}\] To simplify the notations we denote in the following by \(K\) its subset \(K'\) satisfying the above properties.

In the following subsubsections, we will construct affine reparametrizations that satisfy the conditions of Proposition 22, and provide an upper bound estimate on their number.

3.4.2 Decomposition \(K\) by red-Bowen balls↩︎

For every \(x\) and \(n>0\), let (note that \(L_{\beta}^u(g)=L_{\beta}^u(g^q)\)) \[\begin{align} &T_{n}(x):=\{0< i\leq n: G^{i}(x)\in L_{\beta}^u(G)\cap H_{\chi}^u(G)\} \end{align}\] and let \[\chi^{+}(x):=\lceil \log \|D_xG\| \rceil,~~\chi(x):=\lceil \log \|D_xG|_{E^u(x)}\| \rceil.\] For \(E_n \subset \{1,\cdots,n\}\) and \(2n\)-integers \(\{\chi^{+}_i,\chi_i\}_{i=1}^{n}\), we define the set \[\label{e46define-SIgma} \begin{align} &\Sigma(E_n,~\{\chi^{+}_i,\chi_i\}_{i=1}^{n}):=\\ &\left\{x\in K\cap \sigma_{\ast}:T_{n}(x)=E_n,~ \chi^{+}(G^{(j-1)}(x))=\chi^{+}_j,~\chi(G^{(j-1)}(x))=\chi_j,~1\leq j\leq n \right\}. \end{align}\tag{21}\] By a direct computation, we know that \(K\) can be covered by at most \((2qA_{g})^{2n}\) number of the sets of the form \(\Sigma(E_n,\{\chi^{+}_i,\chi_i\}_{i=1}^{n})\) (see 7 for the definition of \(A_g\)). Now, we fix a subset \(\Sigma:=\Sigma(E_n,\{\chi^{+}_i,\chi_i\}_{i=1}^{n})\). For each \(n>0\), we consider \[E^{N}_n:=\{k \in [1,n]: \exists \, 0 \leq i \leq N~\text{s.t.}~k+i \in E_n \, \}.\] It follows from the definitions that if \(\Sigma(E_n, \{\chi^{+}_i,\chi_i\}_{i=1}^{n})\neq \emptyset ~\text{for some}~\big(\{\chi^{+}_i,\chi_i\}_{i=1}^{n}\big)\) then there is a point \(x\in K\) such that \(E_n^N=\{0\leq i<n : G^{i}(x)\in \Lambda\}\). Then, by property (2) of Subsection 3.4.1 we have \[\label{eq:time} \begin{align} \lim_{n\rightarrow \infty} \left(\sup\Big \{\frac{1}{n}\cdot \big| \# E^{N}_n-n\cdot \nu(\Lambda) \big|:~\Sigma(E_n, \{\chi^{+}_i,\chi_i\}_{i=1}^{n})\neq \emptyset ~\text{for some}~\left(\{\chi^{+}_i,\chi_i\}_{i=1}^{n}\right) \Big \}\right)=0 \end{align}\tag{22}\] For convenience, we denote \(E^{N}_n\) as the union of pairwise disjoint intervals⁵ \[E^{N}_n:=\bigcup_{j=1}^{m} (a_j,b_j]:~~0\leq a_j<b_j\leq n,~b_j-a_j\geq N,~b_j\in E_n,~\forall j=1,\cdots,m,~~m\leq \frac{n}{N}+1.\] We define the red-\((n,\varepsilon,G)\)-Bowen balls⁶ with respect to the set \(\Sigma\) by \[\label{e46red-bowen-ball} B^{\rm red}_{G}(x,n,\varepsilon):=\{y \in\Sigma :d(G^{j}(x),G^{j}(y))\leq \varepsilon,~j\in E^{N}_n \}.\tag{23}\] Then, the set \(\Sigma\) can be covered by at most \[\label{e46number-red} \exp \left( \# E^{N}_n \cdot q \big(h_{\rm top}(f)+\alpha \big) \right)\tag{24}\] red-\((n,\varepsilon,G)\)-Bowen balls. Indeed, each interval \((a_j,b_j]\) requires at most \(r(G,b_j-a_j,\varepsilon)\) numbers of \((b_j-a_j,\varepsilon)\)-Bowen balls to cover the whole manifold. It follows from the choice of \(N\) in 11 that \[r\left(G,b_j-a_j,\frac{\varepsilon}{2}\right)\leq \exp \left( (b_j-a_j) \cdot q(h_{\rm top}(f)+\alpha)\right).\] Therefore the total number of red-\((n,\varepsilon,G)\)-Bowen balls needed to cover \(\Sigma\) is bounded above by \[\exp \left( \sum_{j=1}^{m}(b_j-a_j) q\cdot (h_{\rm top}(f)+\alpha) \right) \leq \exp \left( \# E^{N}_n \cdot q(h_{\rm top}(f)+\alpha) \right).\]

3.4.3 Reparametrizations for red-Bowen balls↩︎

Red-Bowen balls are larger than the usual Bowen balls. However, one can still control the number of parametrizations required to cover red-Bowen balls, as in Proposition 22. Recall that \(\sigma\) is a strongly \(\varepsilon\)-bounded curve whose image is contained in \(W^u_{\rm loc}(x_0)\), and that \(\varepsilon\) satisfies \(10 \cdot \varepsilon \leq \varepsilon_{\Upsilon}\), where \(\varepsilon_{\Upsilon}\) satisfies the conclusion of Lemma 16 and \(\|D^jG\|_{\sup}<\Upsilon\) for any \(1\leq j \leq r\).

3.4.4 End of the proof of Proposition 21↩︎

Consider an ergodic measure \(\nu\) as in Proposition 21. Now we give the bound of the numbers of affine reparametrizations.

By Proposition 22 and Proposition 24, we complete the proof of Proposition 21.

4 Proof of Main Theorems: 3-dimensional vector fields↩︎

In this section, we always assume that \(M\) is a three-dimensional compact Remiannian manifold without boundary. Let \(X: M\rightarrow TM\) be a \(C^{\infty}\) vector field without singularities, and let \((\varphi^t)_{t\in \mathbb{R}}\) denote the flow generated by \(X\).

Let \(f:=\varphi^1\) be the time one map of the flow \((\varphi^t)_{t\in \mathbb{R}}\). Since \(X\) has no singularities, the flow direction is uniformly bounded in the following sense: there exists a constant \(C_{X}>0\) such that \[\|D_{x}f^{\pm n}|_{E^c(x)}\|\leq C_{X}, ~\forall n\in \mathbb{N},~\forall x\in M,\] where \(E^c(x)={\rm span}(X(x))\). Therefore, for every hyperbolic measure \(\mu\) of \((\varphi^t)_{t\in \mathbb{R}}\), one has \[\label{eq:flow-Non} \mu\big(\{ x: C_X^{-1}\leq {\rm m}(D_{x}f^n|_{E^c(x)}) \leq \| D_{x}f^n|_{E^c(x)} \| \leq C_X, \forall n\in {\mathbb{Z}}\} \big)=1.\tag{25}\]

Take \(\frac{1}{2}<\alpha_2<\alpha_1<1\) and \(0<\chi<\chi(\alpha_1,\alpha_2)\) (see 3 for the definition of \(\chi(\alpha_1,\alpha_2)\)). By Ruelle’s inequality [19], each ergodic measure \(\mu\) of the flow with positive entropy has exactly one positive and one negative Lyapunov exponents. By Theorem 4, there are \(\beta>0\) and \(C>0\) such that for every ergodic measure \(\nu'\) of \(\varphi^{1}\) with \(h_{\nu'}(\varphi^1)\geq \alpha_1 h_{\rm top}(X)\), one has (see Definition 1 and Definition 2 for notations) \[\nu'(L^u_{\beta}(\varphi^1)\cap L^s_{\beta}(\varphi^1) \cap \mathcal{H}_{C}^{\chi}(\varphi^1))> 2\alpha_2-1>0.\] Recall the Definition 9 of the set \(\mathcal{H}_{\ell}^{\chi,\varepsilon}(X)\). For every \(\varepsilon>0\), by the angel estimate (see Lemma 20) and 25 , one can choose \(\ell > \max\{C,~C_X\}\) such that \(\mathcal{H}_{C}^{\chi}(\varphi^1) \subset \mathcal{H}_{\ell}^{\chi,\varepsilon}(X)\).

Acknowledgement↩︎

The authors gratefully acknowledge the support of the Tianyuan Mathematical Center in Northeast China and thank J. Buzzi and O. Sarig for their helpful comments.

LAMFA, Université Jules Vernes, 80000 Amiens, France.
E-mail: david.burguet@u-picardie.fr

School of Mathematics and Statistics, Jiangxi Normal University, 330022 Nanchang, P. R. China.
E-mail: luochiyi98@gmail.com

School of Mathematical Sciences, Soochow University, 215006 Suzhou, P. R. China.
E-mail: yangdw@suda.edu.cn

References↩︎

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1. D. Yang was partially supported by National Key R&D Program of China (2022YFA1005801), NSFC 12171348 & NSFC 12325106, ZXL2024386 and Jiangsu Specially Appointed Professorship. C. Luo was partially supported by NSFC 12501244.↩︎

2. This gives uniform lower bounds on the sizes of stable and unstable manifolds, independent of the measures.↩︎

3. In general, the definition of Pesin sets is usually given in the non-uniformly hyperbolic setting, where \(E^{c} = \{0\}\). However, to make it easier to study flows, we keep the central direction. In subsequent applications, \(E^c\) primarily corresponds to the flow direction.↩︎

4. By ergodicity we can require equivalently that \(\mu_1(\Lambda_1)=1\) and \(\mu_2(\Lambda_2)=1\).↩︎

5. The last interval \((a_m,b_m]\) may happens \(b_m=n \notin E_n\) and \(b_m-a_m<N\), but this does not affect the final asymptotic estimate of the number of affine reparametrizations, so we omit this case.↩︎

6. In [18], “red" refers to a geometric time, that is, a time at which the point has a long unstable manifold and the derivative satisfies the Pliss condition.↩︎

7. For the initial step \(k=1\), we will construct affine reparametrizations for \(\sigma_0=\sigma\).↩︎