we show that $\int_M \frac{\varphi}{\|\varphi\|} d\mu_{\psi(n)}$ is a Cauchy sequence, and therefore converges towards a limit, denoted by $\int_M \frac{\varphi}{\|\varphi\|} d\mu_{\infty}$. Thus, for every $\varphi \in C_c(M)$, we obtain that

$$ \int_M \varphi d\mu_{\psi(n)} = \|\varphi\| \int_M \frac{\varphi}{\|\varphi\|} d\mu_{\psi(n)} \to \|\varphi\| \int_M \frac{\varphi}{\|\varphi\|} d\mu_{\infty} = \int_M \varphi d\mu_{\infty}. $$

If every $\mu_n$ is $\varphi$-invariant, then the limit $\mu_{\infty}$ of the converging subsequence will be $\varphi$-invariant. Moreover, we can check that $\mu_{\infty}(M) \le 1$. Therefore, $(m_{K_0,L})_{L>0}$ has at least one accumulation point, and all its accumulation points are $\varphi$-invariant and of mass at most 1.

**Proposition 6.1.** Let $\varphi: M \to M$ be a H-flow being $h_{\text{Gur}}$-strongly positive recurrent. Then for every compact set $K_0$ big enough, any accumulation point $m_{\infty}$ of the family $(m_{K_0,L})_{L>0}$ when $L \to +\infty$ is a nonzero finite invariant measure.

*Proof.* Fix a compact set $K_0$ and $\varepsilon > 0$ such that (22) is satisfied. That is

$$ \limsup_{L \to \infty} \frac{1}{L} \log \# \mathcal{P}_{K_0}^{\varepsilon}(L, L + 5\tau_{K_0}) < \lim_{L \to \infty} \frac{1}{L} \log \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}). $$

Let $L_n \to \infty$ be a sequence such that $(m_{K_0,L_n})_n$ converges to $m_{\infty}$ in the vague topology. Let us show that

$$ \liminf_{L \to \infty} m_{K_0,L}(K_0) \ge \varepsilon. $$

Split $\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$ into

$$ \mathcal{P}_{K_0}^{\text{good}}(L, L + 5\tau_{K_0}) = \{\gamma \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}), \ell(\gamma \cap K_0) \ge \varepsilon\ell(\gamma)\} $$

and

$$ \mathcal{P}_{K_0}^{\text{bad}}(L, L + 5\tau_{K_0}) = \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}) \setminus \mathcal{P}_{K_0}^{\text{good}}(L, L + 5\tau_{K_0}) = \mathcal{P}_{K_0}^{\varepsilon}(L, L + 5\tau_{K_0}). $$

Inequality (22) ensures that $\limsup_{L \to \infty} \frac{1}{L} \log \left( \frac{\# \mathcal{P}_{K_0}^{\text{bad}}(L, L + 5\tau_{K_0})}{\# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \right) < 0$, so that

$$ \lim_{L \to +\infty} \frac{\# \mathcal{P}_{K_0}^{\text{bad}}(L, L + 5\tau_{K_0})}{\# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} = 0. \qquad (24) $$

We deduce easily that

$$ \begin{aligned} m_{K_0,L}(K_0) &= \frac{1}{\# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \left( \sum_{\gamma \in \mathcal{P}_{K_0}^{\text{good}}(L, L + 5\tau_{K_0})} \frac{\ell(\gamma \cap K_0)}{\ell(\gamma)} + \sum_{\gamma \in \mathcal{P}_{K_0}^{\text{bad}}(L, L + 5\tau_{K_0})} \frac{\ell(\gamma \cap K_0)}{\ell(\gamma)} \right) \\ &\ge \varepsilon \frac{\# \mathcal{P}_{K_0}^{\text{good}}(L, L + 5\tau_{K_0})}{\# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})}; \end{aligned} $$

thus, we obtain

$$ \liminf_{L \to +\infty} m_{K_0,L}(K_0) \ge \varepsilon. $$

For every continuous map with compact support $\varphi \ge 1_{K_0}$, by definition of vague convergence, we get $\int \varphi dm_{\infty} = \lim_{n \to \infty} \int \varphi dm_{K_0,L_n} \ge \liminf_{n \to \infty} m_{K_0,L_n}(K_0) \ge \varepsilon$. Choosing a decreasing sequence $\varphi_k$ of such maps, with $\lim_{k \to \infty} \varphi_k = 1_{K_0}$, and using the decreasing version of the monotone convergence theorem we obtain $m_{\infty}(K) = \lim_{L \to \infty} \varphi_k dm_{\infty} \ge \varepsilon$. The proposition follows. $\square$

**Notation 6.2 (Measures).** From now, we denote by $m_{\infty}$ an arbitrary fixed accumulation point of the family $(m_{K_0,L})_{L>0}$, and by $m_{\max}$ the invariant probability measure obtained by renormalizing $m_{\infty}$.

**Remark 6.3.** The $h_{\text{Gur}}$-strongly positive recurrent assumption $h_{\text{Gur}}^{\infty}(\varphi) < h_{\text{Gur}}(\varphi)$ is natural, and leads to many natural examples, as shown in [ST21, GST23] in the case of geodesic flows, or in [FSV25]. Moreover, it is stable under perturbations on compact sets.
## 6.2 The subadditivity statements

We will need rigorous versions of the following heuristic statements.

First, given two chords of respective lengths $T_0$ and $T_1$, we can concatenate them, and use the closing lemma to get a periodic orbit of length roughly $T_0 + T_1$. This process is essentially injective: we obtain a lower bound on the number of periodic orbits of length $T_0 + T_1$ in terms of the number of chords of lengths $T_0$ and $T_1$. This lower bound is clearly still valid when we add a linear term $T_0 + T_1$ on the right. This is properly stated in Proposition 6.4.

Second, we wish to say that a periodic orbit of length $T_0 + T_1$ can be divided in two pieces of orbits of respective lengths $T_0$ and $T_1$, and therefore, bound the number of periodic orbits of length $T_0 + T_1$ in terms of the numbers of chords of lengths $T_0$ and $T_1$, respectively. This is much more subtle, for two deep reasons. First, due to the lack of compactness, there is absolutely no reason that an arbitrary periodic orbit return in a compact set at the time $T_0$. The measure constructed in definition 6.2 in section 6.1 will be crucial in the argument, and allow us to say that “most”, or more precisely a positive proportion, of periodic orbits of length $T_0 + T_1$ come back to $K_0$. The second difficulty is that we need a strong subadditivity, with a linear term on the left: $T_0 + T_1$ times the number of periodic orbits of length $T_0 + T_1$ should be smaller than the number of chords of length $T_0$ times the number of chords of length $T_1$. This requires precise statements on the number of returns of a typical periodic orbits, see Lemmas 6.7 and 6.8 and Proposition 6.5.

**Proposition 6.4 (Easy subadditivity).** Let $\varphi: M \to M$ be a H-flow. Let $K_0$ be a compact set with nonempty interior. For every compact set $K \supset K_0$, every $0 < \delta < 1$ and every $0 < \eta < 1$, there exist $T_{\min} > 0$, $S_1 > 0$ and $D_1 > 0$ such that for every $S \ge S_1$, all $x_0, y_0, z_0$ in $K$, and all $T_0, T_1 \ge T_{\min}$, we have:

$$ \mathcal{N}_C(x_0, y_0, \eta, T_0, T_0 + 5\tau_{K_0}, \delta) \times \mathcal{N}_C(y_0, z_0, \eta, T_1, T_1 + 5\tau_{K_0}, \delta) \le D_1 \times (T_0 + T_1 + S) \times \#\mathcal{P}_{K_0}(T_0 + T_1 + S, T_0 + T_1 + S + 5\tau_{K_0}). $$

The following proposition is more difficult. It is stated under the assumption of existence of a nonzero measure $m_{\max}$ as in definition 6.2. This assumption is satisfied as soon as the flow is $h_{\text{Gur}}$-strongly positive recurrent, i.e., $h_{\text{Gur}}^{\infty}(\varphi) < h_{\text{Gur}}(\varphi)$, by Proposition 6.1.

**Proposition 6.5 (Hard subadditivity).** Let $\varphi: M \to M$ be a H-flow that is $h_{\text{Gur}}$-strongly positive recurrent. Let $K_0$ be a compact set with nonempty interior that satisfies inequality (24). Let $L_n \to +\infty$ be a sequence such that the sequence of measures $(m_{K_0, L_n})_n$ converges in the vague topology to a non-zero measure $m_{\infty}$. Let $K$ be a compact subset such that $\mathring{K} \supset K_0$ and $m_{\infty}(\mathring{K}) \ge \frac{3}{4}m_{\infty}(M)$. There exists $\varepsilon_1 > 0$ such that, for all $0 < \eta < \delta < \frac{\varepsilon_1}{4}$ there exist constants $S_2 > 0$ and $D_2 > 0$ such that for every $T > 5\tau_{K_0}$, there exists $k_0 \in \mathbb{N}$, such that for every integer $n \ge k_0$, for all quadruples of points $x_0, y_0, x_1$ and $y_1$ in $K$, and every $S \ge S_2$, we have

$$ L_n \times \#\mathcal{P}_{K_0}(L_n, L_n + 5\tau_{K_0}) \le D_2 \times \mathcal{N}_C(x_0, y_0, \eta, T + S, T + S + 5\tau_{K_0}, \delta) \times \mathcal{N}_C(x_1, y_1, \eta, L_n - T + S, L_n - T + S + 5\tau_{K_0}, \delta). $$

**Remark 6.6.** We could have considered periodic orbits of periods $T_0$ and $T_1$, and stated a variant of the above propositions involving respectively $\#\mathcal{P}_{K_0}(T_0, T_0 + 5\tau_{K_0})$ and $\#\mathcal{P}_{K_0}(T_1, T_1 + 5\tau_{K_0})$ instead of the numbers of chords on the left side of the inequality of Proposition 6.4, and $\mathcal{P}_{K_0}(T_0 + S, T_0 + S + 5\tau_{K_0})$, $\mathcal{P}_{K_0}(L_n - T_0 + S, L_n - T_0 + S + 5\tau_{K_0})$ on the right side of the inequality of Proposition 6.4.

## 6.3 Proof of Proposition 6.4

**Step 1.** Choice of appropriate parameters and notations.

Let $K, K_0, \delta$ and $\eta$ be as in the statement of the Proposition. Let $K_1 = \overline{B(K, 1)}$. The proof will follow from the application of Lemma 2.16 with parameters $K_0 \subset K_1$, with $\nu = \tau_{K_0}/2$, $\delta/4$ and $N = 2$. This lemma gives constants $\sigma > 0$ and $T_{\min} > 0$. Set $S_1 = 2\sigma + 10\tau_{K_0}$. Let $x_0, y_0$ and $z_0$
be in $K$. Let $T_0, T_1 \ge T_{\min}$. Let $E_0$ and $E_1$ be respectively a $E(x_0, y_0, \eta, T_0, T_0 + 5\tau_{K_0}, \delta)$-set and a $E(y_0, z_0, \eta, T_1, T_1 + 5\tau_{K_0}, \delta)$-set of maximal cardinality. In particular

$$ \#E_0 = \mathcal{N}_C(x_0, y_0, \eta, T_0, T_0 + 5\tau_{K_0}, \delta) \quad \text{and} \quad \#E_1 = \mathcal{N}_C(y_0, z_0, \eta, T_1, T_1 + 5\tau_{K_0}, \delta). $$

Throughout the proof, we will see the elements in $E_0$ and $E_1$ as points in $B(x_0, \eta)$ and $B(y_0, \eta)$ or chords with their initial points in $B(x_0, \eta)$ and $B(y_0, \eta)$. In particular, the initial and final points of every chord is in $K_1$, since $\eta < 1$.

**Step 2.** Construction of a map $f$ from chords to periodic orbits.

As $T_0 \ge T_{\min}$ and $T_1 \ge T_{\min}$, for every pair of chords $\beta_0 \in E_0$ and $\beta_1 \in E_1$ with lengths $\ell(\beta_0) \ge T_0$ and $\ell(\beta_1) \ge T_1$, and every $\hat{S} \ge \sigma$, Lemma 2.16 provides a periodic orbit $\gamma$ with length in

$$ \left[ \ell(\beta_0) + \ell(\beta_1) + 2\hat{S} - \tau_{K_1} - \frac{\tau_{K_0}}{2}, \ell(\beta_0) + \ell(\beta_1) + 2\hat{S} + \tau_{K_1} + \frac{\tau_{K_0}}{2} \right] $$

that intersects the interior of $K_0$. Recall that, since $K_0 \subset K_1$, we have $\tau_{K_1} \le \tau_{K_0}$. Thus, the periodic orbit $\gamma$ has length in

$$ \left[ \ell(\beta_0) + \ell(\beta_1) + 2\hat{S} - \frac{3}{2}\tau_{K_0}, \ell(\beta_0) + \ell(\beta_1) + 2\hat{S} + \frac{3}{2}\tau_{K_0} \right]. $$

In particular, for

$$ \hat{S} = \frac{1}{2} \left( S + (T_0 - \ell(\beta_0)) + (T_1 - \ell(\beta_1)) + \frac{3}{2}\tau_{K_0} \right) $$

(notice that $\hat{S} \ge \sigma$ as $S \ge S_1 = 2\sigma + 10\tau_{K_0}$, $T_0 - \ell(\beta_0) \ge -5\tau_{K_0}$ and $T_1 - \ell(\beta_1) \ge -5\tau_{K_0}$) we have

$$ T_0 + T_1 + S \le \ell(\beta_0) + \ell(\beta_1) + 2\hat{S} - \frac{3}{2}\tau_{K_0} $$

and

$$ \ell(\beta_0) + \ell(\beta_1) + 2\hat{S} + \frac{3}{2}\tau_{K_0} \le T_0 + T_1 + S + 5\tau_{K_0}. $$

Then $\gamma \in \mathcal{P}_{K_0}(T_0 + T_1 + S, T_0 + T_1 + S + 5\tau_{K_0})$.

Thus for every $S \ge S_1$, the above construction defines a map

$$ f: E_0 \times E_1 \to \mathcal{P}_{K_0}(T_0 + T_1 + S, T_0 + T_1 + S + 5\tau_{K_0}). $$

Observe that the above construction gives a parametrization of $\gamma$ with an origin $s_0$. More precisely, by Lemma 2.16, there exists $s_0 \in \mathbb{R}$ and $\tau \in [\hat{S} - \tau_{K_1}, \hat{S} + \tau_{K_1}]$ such that,

* for all $s \in [0, \ell(\beta_0)]$, we have $d(\gamma(s_0 + s), \varphi_s(\beta_0)) < \frac{\delta}{4}$,

* for all $s \in [0, \ell(\beta_1)]$, we have $d(\gamma(s_0 + s + \ell(\beta_0) + \tau), \varphi_s(\beta_1)) < \frac{\delta}{4}$.

**Step 3.** Bound on $f^{-1}(\gamma)$.

Consider $\beta_0, \beta'_0 \in E_0, \beta_1, \beta'_1 \in E_1$ such that $(\beta_0, \beta_1) \ne (\beta'_0, \beta'_1)$ and $f(\beta_0, \beta_1) = f(\beta'_0, \beta'_1) = \gamma$. Let $s_0$ (resp. $s'_0$) be the origin of $\gamma$ from the construction $\gamma = f(\beta_0, \beta_1)$ (resp. $\gamma = f(\beta'_0, \beta'_1)$). Assume as a first case that $\beta_0 \ne \beta'_0$. Since they belong to a $E(x_0, y_0, \eta, T_0, T_0 + 5\tau_{K_0}, \delta)$-set, there exist $0 \le u \le T_0 \le \min(\ell(\beta_0), \ell(\beta_1))$, such that

$$ d(\varphi_u(\beta_0), \varphi_u(\beta'_0)) \ge \delta. $$

However, by construction, $\gamma$ satisfies

$$ d(\gamma(s_0 + u), \varphi_u(\beta_0)) \le \delta/4 \quad \text{and} \quad d(\gamma(s'_0 + u), \varphi_u(\beta'_0)) \le \delta/4. $$

Therefore

$$ d(\gamma(s_0 + u), \gamma(s'_0 + u)) \ge \delta/2. $$
As the flow satisfies (2), we have

$$d(\gamma(s_0 + u), \gamma(s'_0 + u)) \leq b|s'_0 - s_0|,$$

so that

$$|s'_0 - s_0| \geq \frac{\delta}{2b}.$$

As the length of $\gamma$ is at most $T_0 + T_1 + S + 5\tau_{K_0}$, it follows that

$$\#\{(\beta'_0, \beta'_1), f(\beta'_0, \beta'_1) = f(\beta_0, \beta_1) \text{ and } \beta'_0 \neq \beta'_1\} \leq (T_0 + T_1 + S + 5\tau_{K_0}) \times \frac{2b}{\delta}.$$

Assume now that $\beta_0 = \beta'_0$ and $\beta_1 \neq \beta'_1$. The construction of the periodic orbit $\gamma$ gives us two constants $\tau$ and $\tau'$ in $[\hat{S} - \tau_{K_0}, \hat{S} + \tau_{K_0}]$ and $[\hat{S}' - \tau_{K_0}, \hat{S}' + \tau_{K_0}]$. Observe that, as $|\hat{S} - \hat{S}'| = \frac{1}{2}|\ell(\beta_1) - \ell(\beta'_1)| \leq 5\tau_{K_0}/2$, we have $|\tau' - \tau| \leq 5\tau_{K_0}$.

Assume that $|\tau' - \tau| \leq \frac{\delta}{4b}$. Since $E_1$ is a $E(y_0, z_0, \eta, T_1, T_1 + 5\tau_{K_0}, \delta)$-set, there exists $0 \leq u \leq T_1$ such that

$$d(\varphi_u(\beta_1), \varphi_u(\beta'_1)) \geq \delta.$$

Arguing as in the first case, we obtain that

$$d(\gamma(s_0 + \ell(\beta_0) + \tau + u), \gamma(s'_0 + \ell(\beta_0) + \tau' + u)) \geq \frac{\delta}{2},$$

and so, using (2), we get

$$d(\gamma(s_0 + \ell(\beta_0) + \tau + u), \gamma(s'_0 + \ell(\beta_0) + \tau + u)) \geq \frac{\delta}{2} - b|\tau' - \tau| \geq \frac{\delta}{4}.$$

Using again (2), we deduce that

$$|s'_0 - s_0| \geq \frac{\delta}{4b}.$$

We then conclude that

$$\#\{(\beta'_0, \beta'_1), f(\beta'_0, \beta'_1) = f(\beta_0, \beta_1) \text{ and } \beta'_0 \neq \beta'_1\} \leq (T_0 + T_1 + S + 5\tau_{K_0}) \times \frac{4b}{\delta} \times 5\tau_{K_0} \times \frac{4b}{\delta}.$$

Thus

$$\#f^{-1}(\gamma) \leq (T_0 + T_1 + S + 5\tau_{K_0}) \times \left(\frac{2b}{\delta} + 5\tau_{K_0} \left(\frac{4b}{\delta}\right)^2\right).$$

By choosing $D_1 > 0$ such that

$$(T_0 + T_1 + S + 5\tau_{K_0}) \times \left(\frac{2b}{\delta} + 5\tau_{K_0} \left(\frac{4b}{\delta}\right)^2\right) \leq D_1(T_0 + T_1 + S),$$

this concludes the proof of Proposition 6.4.

## 6.4 Proof of Proposition 6.5

### 6.4.1 Strategy of the proof

As usual, we want to construct an almost injective map, specifically a map from periodic orbits of period $L_n$ intersecting $K_0$ to chords of length (almost) $T$ and endpoints $x_0, y_0$ and chords of length (almost) $L_n - T$ and endpoints $x_1, y_1$.

For every periodic orbit of length $L$ that intersects $K_0$, the naive idea is to consider a piece of orbit of length $T$ starting and ending in $K_0$ and to cut the periodic orbit at the beginning and the end of this piece of orbit. We then obtain two arcs, one of length $T$ and one of length $L - T$. Using transitivity and the shadowing property, we obtain arcs from $x_0$ to $y_0$ and from $x_1$ to $y_1$.

Unfortunately, there is no guarantee that such arcs exist. The difficulty of the proof consists in finding enough periodic orbits on which enough points of $K_0$ return to $K_0$ after a time $T$. To find
these arcs, we will choose a compact set $K \supset K_0$ with large measure $m_{\max}(K) > 3/4$ so that for every $T > 0$, $m_{\max}(K \cap \varphi_{-T}(K)) > 1/2$. Set $A = K \cap \varphi_{-T}(K)$. We want to find enough intersections between $A$ and $\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$.

As $m_{\infty}(A) > 0$, for $L$ large enough, a positive proportion of periodic orbits in $\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$ spend a positive proportion of their time in $A$. This is proved in Lemma 6.7.

In Lemma 6.8, we prove that if $\mu_{\gamma}(A) \ge \alpha' > 0$ then we control the number of points in $A$ for some appropriate discretization of $\gamma$ of length $T_0$.

Lemmas 6.7 and 6.8 give us Proposition 6.9 where we prove that there are indeed enough periodic orbits with enough point in $K_0$.

### 6.4.2 How often a typical periodic orbit comes back

The existence of a nonzero measure $m_{\max}$ leads to the first important lemma.

**Lemma 6.7.** Let $\varphi: M \to M$ be a flow. Let $K_0$ be a compact set with nonempty interior. Let $A$ be a Borel set, $L > 0$ and $\alpha > 0$ such that $m_{K_0,L}(A) \ge \alpha > 0$. For every $0 < \alpha' < \alpha$, we have

$$ \# \{ \gamma \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}) : \mu_{\gamma}(A) \ge \alpha' \} \ge \frac{\alpha - \alpha'}{1 - \alpha'} \times \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}). $$

*Proof.* Let $n = \# \{ \gamma \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}), \mu_{\gamma}(A) < \alpha' \}$. Then

$$ 0 < \alpha \le m_{K_0,L}(A) \le \frac{\alpha'n + \#\mathcal{P}_{K_0}(L, L + 5\tau_{K_0}) - n}{\#\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})}. $$

Therefore

$$ \alpha \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}) \le \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}) - (1 - \alpha')n. $$

so that $n \le \frac{1-\alpha}{1-\alpha'} \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$. The result follows. $\square$

**Lemma 6.8.** Let $\varphi: M \to M$ be a flow. Let $K_0$ be a compact set with nonempty interior. Let $A$ be a Borel set and $\alpha' > 0$. For all $L > T_0 > 5\tau_{K_0} > 0$, for every periodic orbit $\gamma \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$ such that $\mu_{\gamma}(A) \ge \alpha'$, for any parametrization $\gamma: [0, \ell(\gamma)] \to M$ of $\gamma$ there exists a real number $s \in [0, T_0]$ such that

$$ \# \left\{ i \in \left\{ 0, \dots, \left\lfloor \frac{L}{T_0} \right\rfloor - 1 \right\} , \gamma(s + iT_0) \in A \right\} \ge \left\lfloor \alpha' \left\lfloor \frac{L}{T_0} \right\rfloor \right\rfloor - 3. $$

*Proof.* Set $k = \left\lfloor \frac{L}{T_0} \right\rfloor$ and $k' = \left\lfloor \alpha' \left\lfloor \frac{L}{T_0} \right\rfloor \right\rfloor - 3$. Set $L = kT_0 + r$ with $0 \le r < T_0$. By assumption, since $\mu_{\gamma}(A) \ge \alpha'$, we have $\ell(\gamma \cap A) \ge \alpha'\ell(\gamma)$. As $L \le \ell(\gamma) \le L + 5\tau_{K_0}$, we deduce that

$$ \text{Leb}(\{u \in [0, kT_0], \gamma(u) \in A\}) \ge \alpha'\ell(\gamma) - r - 5\tau_{K_0} \ge \alpha'kT_0 - T_0 - 5\tau_{K_0}. $$

For every $s \in [0, T_0]$, let $J(s) = \{0 \le i \le \left\lfloor \frac{L}{T_0} \right\rfloor - 1, \gamma(s + iT_0) \in A\}$. Suppose that for every $s \in [0, T_0]$, $\#J(s) \le l$, for some integer $0 \le l \le k$.

We now prove that

$$ \text{Leb}(\{u \in [0, kT_0], \gamma(u) \in A\}) \le lT_0. \qquad (25) $$

Write $u = s + iT_0$, with $s \in [0, T_0]$ and $0 \le i \le k - 1$. We have

$$ \begin{aligned} \{u \in [0, kT_0], \gamma(u) \in A\} &= \bigsqcup_{i=0}^{k-1} \{s + iT_0, s \in [0, T_0], \gamma(s + iT_0) \in A\} \\ &= \bigsqcup_{J \in \mathcal{P}(\{0, \dots, k-1\})} (\{s \in [0, T_0], J(s) = J\} + JT_0). \end{aligned} $$

As, for all $J \in \mathcal{P}(\{0, \dots, k-1\})$, we have

$$ \text{Leb}(\{s \in [0, T_0], J(s) = J\} + JT_0) = \#J \times \text{Leb}(\{s \in [0, T_0], J(s) = J\}), $$
as the subsets {$s \in [0, T_0], J(s) = J$} form a measurable partition of $[0, T_0]$ and as $\#J(s) \le l$, the inequality (25) follows.

Therefore, $\alpha'kT_0 - T_0 - 5\tau_{K_0} \le lT_0$, and, since $5\tau_{K_0} < T_0$, we have $l \ge \alpha'k - 2$. For $k' = \lfloor \alpha'k \rfloor - 3$, we have $k' < \alpha'k - 2$. Therefore, $J(s)$ is not bounded by $k'$ for all $s$ and there exists $s \in \mathbb{R}$ such that

$$ \#\{i \in \{0, \dots, k-1\}, \gamma(s + iT_0) \in A\} \ge k'. $$

□

In Subsection 3.3.1, we have defined $\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$ as the set of periodic orbits $((\varphi_t(x))_{t \in \mathbb{R}}, T)$, where $(x, T)$ is a periodic point (that is $\varphi_T(x) = x$). Let $\mathcal{P}'_{K_0}(L, L + 5\tau_{K_0})$ be the set of primitive periodic orbits in $\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$, i.e., the set of periodic orbits $((\varphi_t(x))_{t \in \mathbb{R}}, T)$, where $(x, T)$ is a periodic point and $\varphi_t(x) \neq x$ for every $t \in (0, T)$. The following statement will follow from the fact that:

* almost all periodic orbits are simple;

* if $m_{K_0,L}(A) \ge \alpha$, then a positive proportion of periodic orbits $\gamma$ satisfy $\mu_\gamma(A) \ge \alpha'$ (Lemma 6.7);

* if $\mu_\gamma(A) \ge \alpha'$, then there exists an appropriate discretization of $\gamma$ with step $T_0$ having a positive proportion of points in $A$ (Lemma 6.8).

**Proposition 6.9.** Let $\varphi: M \to M$ be a H-flow such that $h_{\text{Gur}}(\varphi) > 0$. Let $K_0$ be a compact set with nonempty interior. Let $A$ be a Borel set, $0 < \alpha' < \alpha$ and $L > 0$ such that $m_{K_0,L}(A) \ge \alpha > 0$. There exists $N > 0$ such that for all $L > T_0 > 5\tau_{K_0}$, with $\frac{L}{T_0} \ge N$, we have

$$ \# \left\{ \gamma \in \mathcal{P}'_{K_0}(L, L + 5\tau_{K_0}), \exists s \in [0, T_0], \# \left\{ i \in \{0, \dots, \lfloor L/T_0 \rfloor - 1\}, \gamma(s + iT_0) \in A \right\} \ge \frac{\alpha'}{2} \frac{L}{T_0} \right\} \\ \ge 0.99 \times \frac{\alpha - \alpha'}{1 - \alpha'} \times \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}). $$

*Proof.* Choose $N$ large enough so that, for $\frac{L}{T_0} \ge N$, we have $k' = \lfloor \alpha' \lfloor \frac{L}{T_0} \rfloor \rfloor - 3 \ge \frac{\alpha'}{2} \frac{L}{T_0}$. First, observe that a non primitive periodic orbit $\gamma$ is a multiple of a primitive periodic one, that has length at most $\ell(\gamma)/2$. It follows that the number of non-simple periodic orbits in $\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$ is bounded above by $\mathcal{P}_{K_0}(\frac{L+5\tau_{K_0}}{2})$. By Corollary 3.11 (whose hypothesis are satisfied because $\varphi$ is a H-flow with $h_{\text{Gur}}(\varphi) > 0$), it follows that

$$ \lim_{L \to +\infty} \frac{\# \mathcal{P}'_{K_0}(L, L + 5\tau_{K_0})}{\# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} = 1. $$

Up to choose a bigger $N$, we can assume that, if $L > NT_0 > N 5\tau_{K_0}$, we have

$$ \frac{\# \mathcal{P}'_{K_0}(L, L + 5\tau_{K_0})}{\# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \ge 1 - 0.01 \times \frac{\alpha - \alpha'}{1 - \alpha'}. $$

Let $n_\gamma(s) = \#\{i \in \{0, \dots, \lfloor L/T_0 \rfloor - 1\}, \gamma(s + iT_0) \in A\}$. Now, by lemmas 6.7 and 6.8, we have the inequalities

$$ \begin{aligned} \# \left\{ \gamma \in \mathcal{P}'_{K_0}(L, L + 5\tau_{K_0}), \exists s \in \mathbb{R}, n_\gamma(s) \ge \frac{\alpha'}{2} \frac{L}{T_0} \right\} &\ge \# \left\{ \gamma \in \mathcal{P}'_{K_0}(L, L + 5\tau_{K_0}), \exists s \in \mathbb{R}, n_\gamma(s) \ge k' \right\} \\ &\ge \# \left\{ \gamma \in \mathcal{P}'_{K_0}(L, L + 5\tau_{K_0}), \mu_\gamma(A) \ge \alpha' \right\} \\ &\ge \# \left\{ \gamma \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}), \mu_\gamma(A) \ge \alpha' \right\} \\ &\quad - 0.01 \times \frac{\alpha - \alpha'}{1 - \alpha'} \times \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}) \\ &\ge \frac{\alpha - \alpha'}{1 - \alpha'} \times \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}) \\ &\quad - 0.01 \times \frac{\alpha - \alpha'}{1 - \alpha'} \times \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}) \\ &\equiv 0.99 \times \frac{\alpha - \alpha'}{1 - \alpha'} \times \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}). \end{aligned} $$

□
### 6.4.3 Proof of Proposition 6.5

Recall that $m_\infty$ is the limit, in the vague topology, of a sequence $(m_{K_0,L_n})_{n \in \mathbb{N}}$ when $L_n \to \infty$, and $m_{\max}$ is the probability measure obtained by renormalization of $m_\infty$.

**Step 1.** Setting the parameters. Choose some large compact set $K$ such that $\tilde{K} \supset K_0$ and $m_{\max}(\tilde{K}) > 3/4$. In particular, for every $T > 0$, we obtain that $m_{\max}(K \cap \varphi_{-T}(K)) > 1/2$. Fix $T > 5\tau_{K_0}$. Let $A = K \cap \varphi_{-T}(K)$. Let $\alpha = \frac{m_\infty(A)}{2} > 0$. In particular, there exists $k_0$ such that for all $n \ge k_0$, we have $m_{K_0,L_n}(A) \ge \alpha$. Choose $0 < \alpha' < \alpha$. Lemma 2.13 applied with $\nu = 1$ and $\tau_1 = 5\tau_{K_0}$ provides us with constants $\tau_0$ and $\varepsilon_1$ which will be used below. Fix $\eta$ and $\delta$ such that $0 < \eta < \delta < \varepsilon_1/4$. Fix $T_0 = 6\tau_{K_0}$.

The finite exact shadowing, i.e., Proposition 2.6, applied with the compact set $\overline{B(K, 1)}$, with $\delta = \frac{\eta}{2}$ and $N = 3$, gives us a constant $0 < \rho < \frac{\eta}{2}$. The uniform transitivity, i.e., Lemma 2.14, applied with the compact set $K$ and with $\delta = \rho$, provides a constant $\sigma \ge 0$. Let $S_2 = 2\sigma + 2\tau_K + 10\tau_{K_0}$. Fix some $S \ge S_2$.

Let $x_0, y_0, x_1, y_1$ be four arbitrary points in $K$. Let $E_0$ be a $E(x_0, y_0, \eta, T+S, T+S+5\tau_{K_0}, \delta)$-set of maximal cardinality and $E_1$ be a $E(x_1, y_1, \eta, L_n - T + S, L_n - T + S + 5\tau_{K_0}, \delta)$-set of maximal cardinality. In particular, we have

$$ \#E_0 = \mathcal{N}_C(x_0, y_0, \eta, T+S, T+S+5\tau_{K_0}, \delta) \quad \text{and} \quad \#E_1 = \mathcal{N}_C(x_1, y_1, \eta, L_n - T+S, L_n - T+S+5\tau_{K_0}, \delta). $$

**Step 2.** Some preliminary estimates on the number of cutting points. Denote by $\tilde{P}_{K_0}(L_n)$ the set

$$ \{ \gamma \in \mathcal{P}'_{K_0}(L_n, L_n + 5\tau_{K_0}), \exists s \in [0, T_0], \#\{i \in [0, \dots, \lfloor L/T_0 \rfloor - 1], \gamma(s + iT_0) \in A\} \ge \frac{\alpha'}{2} \frac{L}{T_0} \}. $$

From Proposition 6.9, since $m_{K_0,L_n}(A) \ge \alpha > 0$, we know that

$$ \#\tilde{P}_{K_0}(L_n) \ge 0.99 \times \frac{\alpha - \alpha'}{1 - \alpha'} \times \#\mathcal{P}_{K_0}(L_n, L_n + 5\tau_{K_0}). $$

Without loss of generality, until the end of the proof, we reparametrize each $\gamma \in \tilde{P}_{K_0}(L_n)$ so that $s = 0$. Let $\mathcal{D}_\gamma$ be the set of times of the form $iT_0$, such that $\gamma(iT_0) \in A$. By the definition of $\tilde{P}_{K_0}(L_n)$, we have

$$ \#\mathcal{D}_\gamma \ge \frac{\alpha'}{2} \times \frac{L_n}{T_0} = \frac{\alpha'}{12\tau_{K_0}} \times L_n. $$

Therefore

$$ \# \bigcap_{\gamma \in \tilde{P}_{K_0}(L_n)} \{\gamma\} \times D_{\gamma} \ge 0.99 \times \frac{\alpha - \alpha'}{1 - \alpha'} \times \frac{\alpha'}{12\tau_{K_0}} \times L_n \times \#\mathcal{P}_{K_0}(L_n, L_n + 5\tau_{K_0}). \quad (26) $$

**Step 3.** A map $F$ from periodic orbits to chords.

The intuitive idea is the following. For each point $\gamma(iT_0) \in A$, as $A = K \cap \varphi_{-T}(K)$, we know that $\varphi_T(\gamma(iT_0)) \in K$. Cutting the orbit $\gamma$ at these two points we get two chords of respective lengths $T$ and $\ell(\gamma) - T \in [L_n - T, L_n - T + 5\tau_{K_0}]$. Using transitivity, we add some small arcs to obtain chords from $x_0$ to $y_0$ and from $x_1$ to $y_1$.

Let $\gamma \in \tilde{P}_{K_0}(L_n)$ and $i \in D_\gamma$. Using transitivity and finite exact shadowing, we can build a chord $\beta^0$ with length $\ell(\beta^0) \in [T+S, T+S+4\tau_K]$ that goes from $B(x_0, \eta)$ at time 0, to $B(\gamma(iT_0), \eta/2)$ at time $t_{x_0} \in [\frac{S}{2}, \frac{S}{2} + 2\tau_K]$, then follows exactly $\gamma$ at a distance at most $\eta/2$ during a time $T$, and then goes from $B(\gamma(iT_0 + T), \eta/2)$ to $B(y_0, \eta)$ at time $\ell(\beta^0)$. For the remaining part of the proof, we denote by $\beta_{x_0}^0$ the restriction of $\beta^0$ to the interval $[0, t_{x_0}]$, $\beta_\gamma^0$ the restriction of $\beta^0$ to the interval $[t_{x_0}, t_{x_0} + T]$, and $\beta_{y_0}^0$ the restriction of $\beta^0$ to the interval $[t_{x_0} + T, \ell(\beta^0)]$.

Similarly, we get a chord $\beta^1$ from $B(x_1, \eta)$ to $B(y_1, \eta)$ of length $\ell(\beta^1) \in [L_n - T + S, L_n - T + S + 4\tau_K]$ that goes from $B(x_1, \eta)$ at time 0, to $B(\gamma(iT_0 + T), \eta/2)$ at some time $t_{x_1} \in [\frac{S}{2}, \frac{S}{2} + 2\tau_K]$, then follows exactly $\gamma$ at a distance at most $\eta/2$ during a time $\ell(\gamma) - T$, and then goes from $B(\gamma(iT_0), \eta/2)$ to
$B(y_1, \eta)$ at a time $\ell(\beta^1) \in [L_n - T + S, L_n - T + S + 4\tau_K]$. Observe that, in such a construction, the last part of the chord is obtained from a path built by transitivity whose length depends on the lengths of the chords of the first part of the construction, similarly to what is done in the proof of Proposition 5.7: this enables to have a more precise control on the length of the final chord. As above, we denote by $\beta_{x_1}^1$ the restriction of $\beta^1$ to $[0, t_{x_1}]$, $\beta_\gamma^1$ the restriction of $\beta^1$ to $[t_{x_1}, t_{x_1} + \ell(\gamma) - T]$ and $\beta_{y_1}^1$ the restriction of $\beta^1$ to $[t_{x_1} + \ell(\gamma) - T, \ell(\beta^1)]$. Now, in order to define the image of $\gamma$ by $F$, we consider the pair of chords $(\bar{\beta}_0, \bar{\beta}_1) \in E_0 \times E_1$ that is the closest to the pair $(\beta^0, \beta^1)$.

Therefore, we obtain a map

$$F: \bigcup_{\gamma \in \tilde{P}_{K_0}(L_n)} \{\gamma\} \times \mathcal{D}_{\gamma} \to E_0 \times E_1.$$

**Step 4.** The map $F$ is almost injective.

Assume that $(\gamma, iT_0)$ and $(\tilde{\gamma}, \tilde{i}T_0)$ lead to the same pair $(\bar{\beta}_0, \bar{\beta}_1) \in E_0 \times E_1$, where $\gamma, \tilde{\gamma} \in \tilde{P}_{K_0}(L_n)$ and $i \in \mathcal{D}_{\gamma}, \tilde{i} \in \mathcal{D}_{\tilde{\gamma}}$. First, assume that

$$|\ell(\gamma) - \ell(\tilde{\gamma})| \le \tau_0, \qquad (27)$$

where $\tau_0$ is defined in Step 1.

The above construction associates to $(\gamma, iT_0)$ (resp. $(\tilde{\gamma}, \tilde{i}T_0)$) two chords divided in three parts $\beta^0 = (\beta_{x_0}^0, \beta_\gamma^0, \beta_{y_0}^0)$ and $\beta^1 = (\beta_{x_1}^1, \beta_\gamma^1, \beta_{y_1}^1)$ (resp. $\tilde{\beta}^0 = (\tilde{\beta}_{x_0}^0, \tilde{\beta}_\gamma^0, \tilde{\beta}_{y_0}^0)$ and $\tilde{\beta}^1 = (\tilde{\beta}_{x_1}^1, \tilde{\beta}_\gamma^1, \tilde{\beta}_{y_1}^1)$) that are $\delta$-close to the chord $\bar{\beta}_0$, during a time at least $T + S$, and to the chord $\bar{\beta}_1$, during a time at least $L_n - T + S$, respectively. Assume also that

$$\max \left\{ \left| \ell(\tilde{\beta}_{x_0}^0) - \ell(\beta_{x_0}^0) \right|, \left| \ell(\tilde{\beta}_{y_0}^0) - \ell(\beta_{y_0}^0) \right|, \left| \ell(\tilde{\beta}_{x_1}^1) - \ell(\beta_{x_1}^1) \right|, \left| \ell(\tilde{\beta}_{y_1}^1) - \ell(\beta_{y_1}^1) \right| \right\} \le \frac{\delta}{b}. \quad (28)$$

By construction, for every $u \in [0, T]$, we have

$$d(\bar{\beta}_0(\ell(\beta_{x_0}^0) + u), \beta^0(\ell(\beta_{x_0}^0) + u)) \le \delta$$

and

$$d(\beta^0(\ell(\beta_{x_0}^0) + u), \gamma(iT_0 + u)) \le \eta/2.$$

Therefore, for every $u \in [0, T]$,

$$d(\bar{\beta}_0(\ell(\beta_{x_0}^0) + u), \gamma(iT_0 + u)) \le \delta + \eta/2.$$

Similarly, for every $u \in [0, T]$,

$$d(\bar{\beta}_0(l(\tilde{\beta}_{x_0}^0) + u), \tilde{\gamma}(iT_0 + u)) \le \delta + \eta/2.$$

As, by (2),

$$d(\bar{\beta}_0(\ell(\beta_{x_0}^0) + u), \bar{\beta}_0(\ell(\tilde{\beta}_{x_0}^0) + u)) \le b \left| \ell(\tilde{\beta}_{x_0}^0) - \ell(\beta_{x_0}^0) \right| \le \delta,$$

for every $u \in [0, T]$, we get, since $\eta < \delta < \frac{\varepsilon_1}{4}$,

$$d(\gamma(iT_0 + u), \tilde{\gamma}(iT_0 + u)) \le 4\delta < \varepsilon_1.$$

On the other part of the orbit, the same reasoning gives for every $u \in [0, L_n - T]$

$$d(\gamma(iT_0 + T + u), \tilde{\gamma}(iT_0 + T + u)) \le 4\delta < \varepsilon_1.$$

Therefore $\gamma$ and $\tilde{\gamma}$ are $\varepsilon_1$-close on an interval of length $L_n$. Lemma 2.13 implies that $\gamma = \tilde{\gamma}$ and $i = \tilde{i}$.

This proves that for every choice of length of $\ell(\gamma) \in [L_n, L_n + 5\tau_{K_0}]$ up to $\tau_0$ as in (27), and every choice respectively of $\ell(\beta_{x_0}^0)$, $\ell(\beta_{y_0}^0)$, $\ell(\beta_{x_1}^1)$, $\ell(\beta_{y_1}^1)$ in $[\frac{S}{2}, \frac{S}{2} + 2\tau_K]$ up to $\delta/b$ as in (28), there is at most one pair $(\gamma, iT_0)$ leading to $(\bar{\beta}^0, \bar{\beta}^1)$. As a consequence, there are at most $\left(\left\lfloor \frac{5\tau_{K_0}}{\tau_0} \right\rfloor + 1\right) \times \left(\left\lfloor \frac{2\tau_K b}{\delta} \right\rfloor + 1\right)^4$
pairs $(\gamma, iT_0)$ with $\gamma$ a periodic orbit in $\tilde{\mathcal{P}}_{K_0}(L_n)$ and $iT_0 \in \mathcal{D}_\gamma$ that lead to the pair of chords $(\bar{\beta}_0, \bar{\beta}_1)$. Therefore

$$ \# \bigcup_{\gamma \in \tilde{\mathcal{P}}_{K_0}(L_n)} (\{\gamma\} \times \mathcal{D}_{\gamma}) \le \left( \lfloor \frac{5\tau_{K_0}}{\tau_0} \rfloor + 1 \right) \times \left( \lfloor \frac{2\tau_K b}{\delta} \rfloor + 1 \right)^4 \times \#E_0 \times \#E_1. \quad (29) $$

**Step 5. Conclusion.**

This allows to conclude the proof of Proposition 6.5. Indeed, by (26) and (29),

$$ \begin{aligned} L_n \times \#\mathcal{P}_{K_0}(L_n, L_n + 5\tau_{K_0}) &\le \frac{1}{0.99} \frac{1-\alpha'}{\alpha-\alpha'} \times \frac{12\tau_{K_0}}{\alpha'} \times \# \bigcup_{\gamma \in \tilde{\mathcal{P}}_{K_0}(L_n)} (\{\gamma\} \times \mathcal{D}_{\gamma}) \\ &\le \frac{1}{0.99} \frac{1-\alpha'}{\alpha-\alpha'} \times \frac{12\tau_{K_0}}{\alpha'} \times \left( \lfloor \frac{5\tau_{K_0}}{\tau_0} \rfloor + 1 \right) \times \left( \lfloor \frac{2\tau_K b}{\delta} \rfloor + 1 \right)^4 \times \#E_0 \times \#E_1 \\ &= D_2 \times \mathcal{N}_C(x_0, y_0, \eta, T+S, T+S+5\tau_{K_0}, \delta) \\ &\quad \times \mathcal{N}_C(x_1, y_1, \eta, L_n-T+S, L_n-T+S+5\tau_{K_0}, \delta) \end{aligned} $$

with

$$ D_2 = \frac{1}{0.99} \frac{1-\alpha'}{\alpha-\alpha'} \times \frac{12\tau_{K_0}}{\alpha'} \times \left( \lfloor \frac{5\tau_{K_0}}{\tau_0} \rfloor + 1 \right) \times \left( \lfloor \frac{2\tau_K b}{\delta} \rfloor + 1 \right)^4. $$

# 7 The measure maximizes the entropy

In this section, $K_0$ is a compact set with nonempty interior as in section 6.1 and $K \supset K_0$ is a larger compact set, such that $K_0 \subset \mathring{K}$.

In the first section, we prove uniform estimates on $m_{K_0,L}(B(x, T, \varepsilon))$ for $x \in K$. In the second section, we take the limit when $L \to +\infty$ and obtain uniform estimates on the measure $m_{\max}(B(x, T, \varepsilon))$, for every $x \in K$ (for every compact set $K \subset M$). In the last section, we finally prove our main theorem.

## 7.1 Relation between $m_{K_0,L}$ and the number of chords

The heuristics of this section is the following. Recall that the measure $m_{K_0,L}$ is defined in (23) as the average of the periodic invariant probability measures $\mu_\gamma$, where $\gamma$ varies over all periodic orbits in $\mathcal{P}_{K_0}(L, L+5\tau_{K_0})$, i.e., those intersecting $K_0$, with length in $[L, L+5\tau_K]$. Given some dynamical ball $B(x, T, \varepsilon)$ for $x \in M$, the measure $m_{K_0,L}(B(x, T, \varepsilon))$ satisfies therefore

$$ \begin{aligned} m_{K_0,L}(B(x, T, \varepsilon)) &= \frac{1}{\#\mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \sum_{\gamma \in \mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \mu_\gamma(B(x, T, \varepsilon)) \\ &\approx \frac{1}{L \times \#\mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \sum_{\gamma \in \mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \ell(\gamma \cap B(x, T, \varepsilon)). \end{aligned} $$

Now, the heart of the argument is the proof that the last sum is comparable to the number $\mathcal{N}_C(\varphi_T(x), x, \eta, L-T, \delta)$ of chords from $\varphi_T(x)$ to $x$, so that

$$ m_{K_0,L}(B(x, T, \varepsilon)) \approx \frac{1}{L \times \#\mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \times \mathcal{N}_C(\varphi_T(x), x, L-T, \delta). $$

The idea of the proof consists in the following remark. Given a chord starting at a point $z$ of length roughly $L-T$ from a neighbourhood of $\varphi_T(x)$ to a neighbourhood of $x$, and using the closing lemma as in Lemma 2.16, we can build a periodic orbit in $\mathcal{P}_{K_0}(L, L+5\tau_K)$ following first $(\varphi_s(x))_{0 \le s \le T}$ and afterwards the chord $(\varphi_s(z))_{0 \le s \le L-T}$, and intersecting $B(x, \varepsilon, T)$. Conversely, given a periodic orbit $\gamma \in \mathcal{P}_{K_0}(L, L+5\tau_K)$ with an origin $w \in B(x, \varepsilon, T)$, we can cut $\gamma$ at 0 and $T$, and get a chord of length roughly $L-T$ from a neighbourhood of $\varphi_T(x)$ to a neighbourhood of $x$. The difficulty of the argument is to show that the above constructions are almost one-to-one, or more precisely, that
each preimage of the corresponding maps between chords and periodic orbits has bounded cardinality. The rigorous details corresponding to the above heuristics are provided in Lemmas 7.3 (lower bound) and 7.5 (upper bound). To this purpose, we will need to notice in Lemma 7.2 that each return of $\gamma$ in $B(x, T, \varepsilon)$ has a bounded length and in Lemma 7.4 that these returns are not too close one from another.

We start with an easy but useful observation. Recall that $b$ is defined in Equation (2).

**Lemma 7.1.** For all $\varepsilon > 0$, $T > 0$, $x \in M$, and $y \in B(x, \frac{\varepsilon}{2}, T)$, for every $0 \le s < \frac{\varepsilon}{2b}$, we have $\varphi_s(y) \in B(x, \varepsilon, T)$.

*Proof.* For every $s \in [0, \frac{\varepsilon}{2b}[$, and every $\tau \in [0, T]$, we have

$$d(\varphi_{\tau+s}(y), \varphi_{\tau}(x)) \le d(\varphi_{\tau+s}(y), \varphi_{\tau}(y)) + d(\varphi_{\tau}(y), \varphi_{\tau}(x)) \le |s|b + \frac{\varepsilon}{2} < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon$$

so that $\varphi_s(y) \in B(x, \varepsilon, T)$, as desired.

□

In Lemmas 7.2 and 7.4, given a set $C$, and a point $y \in C$, we consider the connected component of $0$ in $\{s \in \mathbb{R}, \varphi_s(y) \in C\}$. It is an interval that we denote by $J_C(y) = (J_C^{\min}(y), J_C^{\max}(y))$. In Lemma 7.2, we prove that for a dynamical ball $B(x, \varepsilon, T)$, and any $y \in B(x, \varepsilon, T)$, the size of the interval $J_{B(x,\varepsilon,T)}(y)$ is bounded.

**Lemma 7.2.** Let $K \subset M$ be compact and $\Delta > 0$. There exist $\varepsilon_{K,\Delta} > 0$ such that for every $0 < \varepsilon \le \varepsilon_{K,\Delta}$, $x \in K$, $T \ge 1$ and $y \in B(x, \varepsilon, T)$, we have

$$\mathrm{Leb}\left(J_{B(x,\varepsilon,T)}(y)\right) = J_{B(x,\varepsilon,T)}^{\max}(y) - J_{B(x,\varepsilon,T)}^{\min}(y) \le \mathrm{Leb}\left(J_{B(x,\varepsilon,1)}(y)\right) \le \Delta.$$

*Proof.* First note that for all $T \ge 1$, $x \in M$ and $\varepsilon > 0$, we have $B(x, \varepsilon, T) \subset B(x, \varepsilon, 1)$ and therefore $J_{B(x,\varepsilon,T)}(y) \subset J_{B(x,\varepsilon,1)}(y)$. For every $x \in K$, there exists $\tau_0 = \tau_0(x) > 0$, $r = r(x) > 0$ and a flow-bow $\Omega$ centered at $x$ diffeomorphic to $B(0, r) \times (-\tau_0, \tau_0)$ (where $B(0, r)$ is a ball of radius $r$ in $\mathbb{R}^{\dim(M)-1}$). As $\Omega$ is a neighborhood of $x$, there exists $\varepsilon(x)$ such that $B(x, \varepsilon(x), 1) \subset \Omega$. For every $0 < \varepsilon \le \varepsilon(x)$ and every $y \in B(x, \varepsilon, 1)$, we have $B(x, \varepsilon, 1) \subset B(x, \varepsilon(x), 1) \subset \Omega$. Therefore, $\mathrm{Leb}\left(J_{B(x,\varepsilon,1)}(y)\right) \le 2\tau_0$. We may reduce $\tau_0$ so that $\tau_0 \le \Delta/2$. As $K$ is compact, one can choose $\tau_0$ and $\varepsilon$ uniformly in $x$. This concludes the proof of the lemma.
□

We are now able to prove the first key lemma of this section.

**Lemma 7.3.** Let $K_0$ be a compact set with nonempty interior as in Section 6.1, and $K \supset K_0$ be a larger compact set. For every $\delta > 0$, there exists $\varepsilon_{K,\delta} > 0$ and $T'_{\min} > 1$ such that for every $0 < \varepsilon < \varepsilon_{K,\delta}$, there exist $\sigma = \sigma_{K,\delta,\varepsilon} > 0$ and $\eta_{K,\delta,\varepsilon} > 0$ such that for every $0 < \eta < \eta_{K,\delta,\varepsilon}$, all $L, T$ such that $L \ge T + T'_{\min}$ and $T \ge T'_{\min}$ and $x \in K$ such that $\varphi_T(x) \in K$, we have

$$\frac{\varepsilon}{4b} \times \frac{N_C(\varphi_T(x), x, \eta, L - T - \sigma_{K,\delta,\varepsilon}, L - T - \sigma_{K,\delta,\varepsilon} + 5\tau_{K_0}, \delta)}{L + \#P_{K_0}(L, L + 5\tau_{K_0})} \le m_{K_0,L}(B(x, \varepsilon, T)).$$

*Proof.* The heuristics of the proof is the following. Thanks to Lemma 2.17, we can glue any chord of length roughly $L - T$ starting close to $\varphi_T(x)$ and arriving close to $x$ to the orbit $(\varphi_t(x))_{0 \le t \le T}$ to get a periodic orbit of length $L$ intersecting $B(x, \varepsilon, T)$. Doing it with enough care will guarantee that this intersection point lies inside $B(x, \varepsilon/2, T)$, and therefore the orbit spends a time at least $\frac{\varepsilon}{2b}$ inside $B(x, \varepsilon, T)$. This will allow to get the desired bound. Let us start the rigorous argument.

**Step 1.** Choice of constants. Choose $T'_{\min}$ so that for $L \ge T'_{\min}$, we have $\frac{1}{L+5\tau_{K_0}} \ge \frac{1}{2L}$. Fix $\delta > 0$. Then, Lemma 7.2 applied on $K$ with $\Delta = \delta/2b$ gives us a constant $\varepsilon_{K,\delta} > 0$.

Fix $0 < \varepsilon < \varepsilon_{K,\delta}$ and set $\delta' = \min(\delta/4, \varepsilon/2, 1)$. Lemma 2.17 applied with $K_0 \subset \overline{B(K, 1)}$, $\delta = \delta' > 0$, $\nu = \tau_{K_0}$ and $N = 2$, gives us $\eta_{K_0,\delta,\varepsilon} \le 1$, $\sigma > 0$ and $T_{\min} > 0$ such that the following holds. Let $\sigma_{K,\delta,\varepsilon} := \sigma + 3\tau_{K_0}$. First, increase $T'_{\min}$ so that $T'_{\min} \ge T_{\min}$ and $T'_{\min} \ge \sigma_{K,\delta,\varepsilon}$. Then, for every $0 < \eta < \eta_{K_0,\delta,\varepsilon}$, $L, T$ such that $L \ge T + T'_{\min}$ and $T \ge T'_{\min}$ and any chord $z$ from $B(\varphi_T(x), \eta)$ to $B(x, \eta)$ with length $\ell(z)$ such that

$$L - T - \sigma_{K,\delta,\varepsilon} \le \ell(z) \le L - T - \sigma_{K,\delta,\varepsilon} + 5\tau_{K_0},$$
let

$$S = \sigma_{K,\delta,\varepsilon} - \tau_{K_0} + (L - T - \sigma_{K,\delta,\varepsilon} + 5\tau_{K_0} - \ell(z)) = L - T - \ell(z) + 2\tau_{K_0} \geq \sigma;$$

By Lemma 2.17, there exists a periodic orbit $\gamma$ with length

$$\ell(\gamma) \in [T + \ell(z) + S - 2\tau_{K_0}, T + \ell(z) + S + 2\tau_{K_0}] = [L, L + 4\tau_{K_0}] \subset [L, L + 5\tau_{K_0}]$$

that intersects $\mathring{K}_0$, $\delta'$-shadows first $(\varphi_t(x))_{0\leq t\leq T}$ and then $(\varphi_t(z))_{0\leq t\leq \ell(z)}$. In particular, $\gamma(0) \in B(x, \delta')$. Observe that $\gamma \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$.

**Step 2.** A first lower bound for $m_{K_0,L}(B(x, \varepsilon, T))$. Denote by $\mathcal{J}(L + 5\tau_{K_0})$ the set of intervals included in $[0, L + 5\tau_{K_0}]$. For each periodic orbit $\gamma \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$, choose a parametrization such that $\gamma(0) \notin B(x, \varepsilon, T)$. Denote by $\Theta_{K_0,L}(x, \varepsilon, T)$ the set of pairs $(\gamma, I) \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0}) \times \mathcal{J}(L + 5\tau_{K_0})$ such that for every $\gamma \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$ (with its parametrization and associated origin), and every $s \in I$, $\gamma(s) \in B(x, \varepsilon, T)$. Moreover, assume $I$ is maximal for this property. In a similar way, let $\Theta'_{K_0,L}(x, \varepsilon, T)$ be set of pairs $(\gamma, I) \in \Theta_{K_0,L}(x, \varepsilon, T)$ such that $\gamma(I) \cap B(x, \varepsilon/2, T) \neq \emptyset$.

Observe that for $L \geq T'_{\min}$, we have

$$\begin{aligned}
m_{K_0,L}(B(x, \varepsilon, T)) &= \frac{1}{\#\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \sum_{\gamma \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \mu_{\gamma}(B(x, \varepsilon, T)) \\
&= \frac{1}{\#\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \sum_{(\gamma, I) \in \Theta_{K_0,L}(x, \varepsilon, T)} \frac{\text{Leb}(I)}{\ell(\gamma)} \\
&\geq \frac{1}{\#\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \sum_{(\gamma, I) \in \Theta'_{K_0,L}(x, \varepsilon, T)} \frac{\text{Leb}(I)}{\ell(\gamma)} \\
&\geq \frac{1}{\#\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \sum_{(\gamma, I) \in \Theta'_{K_0,L}(x, \varepsilon, T)} \frac{\text{Leb}(I)}{2L} \\
&\geq \frac{1}{\#\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \times \frac{\varepsilon}{4bL} \times \#\Theta'_{K_0,L}(x, \varepsilon, T).
\end{aligned}$$

where the second lower bound follows from the inequalities $L \leq \ell(\gamma) \leq L + 5\tau_{K_0}$ and $\frac{1}{L + 5\tau_{K_0}} \geq \frac{1}{2L}$, and the third lower bound comes from Lemma 7.1.

**Step 3.** Going from chords to periodic orbits. Let $E$ be a $E(\varphi_T(x), x, \eta, L - T - \sigma_{K,\delta,\varepsilon}, L - T - \sigma_{K,\delta,\varepsilon} + 5\tau_{K_0}, \delta)$-set of maximal cardinality, i.e., so that

$$\#E = \mathcal{N}_c(\varphi_T(x), x, \eta, L - T - \sigma_{K,\delta,\varepsilon}, L - T - \sigma_{K,\delta,\varepsilon} + 5\tau_{K_0}, \delta).$$

Applying Lemma 2.17 as described above, we can associate to each $z$ in $E$ a periodic orbit $\gamma$ in $\mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$. Moreover, we know that the origin $s_0$ given by the construction is such that $\gamma(s_0) \in B(x, \delta', T) \subset B(x, \varepsilon/2, T)$. This point in $B(x, \varepsilon/2, T)$ gives us an interval $J$ with $(\gamma, J) \in \Theta'_{K_0,L}(x, \varepsilon, T)$ and therefore a map

$$\theta: E \to \Theta'_{K_0,L}(x, \varepsilon, T).$$

**Step 4.** The map $\theta$ is injective. Assume that two chords $z_1, z_2$ in $E$ have same image $(\gamma, J)$. For $\gamma$ we use the parametrization from the definition of $\Theta_{K_0,L}(x, \varepsilon, T)$. The origins $s_1$ and $s_2$ associated to the construction of $\theta(z_1)$ and $\theta(z_2)$ satisfy $s_1, s_2 \in J$ and $\gamma(s_1), \gamma(s_2) \in B(x, \varepsilon/2, T)$. By Lemma 7.2, as $\varepsilon \leq \varepsilon_{K,\delta}$, we have $|J| \leq \delta/2b$ (see Step 1). Therefore $|s_1 - s_2| \leq \delta/2b$. An elementary computation gives, for every $0 \leq s \leq \min(\ell(z_1), \ell(z_2))$, where $\ell(z_1), \ell(z_2)$ denote the length of the chords of $z_1, z_2$ respectively,

$$\begin{aligned}
d(\varphi_{s+T+s_1}(z_1), \varphi_{s+T+s_2}(z_2)) &\leq d(\varphi_{s+T+s_1}(z_1), \gamma(s+T+s_1)) + d(\gamma(s+T+s_1), \gamma(s+T+s_2)) \\
&\quad + d(\gamma(s+T+s_2), \varphi_{s+T+s_2}(z_2)) \\
&< \delta' + \delta/2 + \delta' \leq \delta.
\end{aligned}$$

As $E$ is a $(\delta, L - T - \sigma_{K,\delta,\varepsilon})$-separating set, we deduce that $z_1 = z_2$, so that $\theta$ is injective. Therefore

$$\#E \leq \#\Theta'_{K_0,L}(x, \varepsilon, T).$$
**Conclusion.** The above arguments show that

$$ 
\begin{aligned}
m_{K_0,L}(B(x, \varepsilon, T)) &\geq \frac{\varepsilon}{4b} \times \frac{1}{L \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \times \# \Theta'_{K_0,L}(x, \varepsilon, T) \\
&\geq \frac{\varepsilon}{4b} \times \frac{1}{L \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})} \times \# E \\
&= \frac{\varepsilon}{4b} \times \frac{\mathcal{N}_C(\varphi_T(x), x, \eta, L - T - \sigma_{K,\delta,\varepsilon}, L - T - \sigma_{K,\delta,\varepsilon} + 5\tau_{K_0}, \delta)}{L \# \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})}.
\end{aligned}
$$

□

Our next goal is to bound $m_{K_0,L}(B(x, \varepsilon, T))$ from above. We will need to bound from above the total amount of time that a periodic orbit $\gamma \in \mathcal{P}_{K_0}(L, L + 5\tau_{K_0})$ spends in $B(x, \varepsilon, T)$. We know from Lemma 7.2 that each interval of time that such an orbit spends in $B(x, \varepsilon, T)$ has a bounded length. Lemma 7.4 is the second technical lemma. It allows to say that, up to increasing slightly $B(x, \varepsilon, T)$ to make it more smooth, the distance between two such intervals is bounded from below by a uniform constant, which allows to bound as desired the total amount of time in $B(x, \varepsilon, T)$.

**Lemma 7.4** (No immediate return). Let $K \subset M$ be a compact set. There exist $\Delta_K > 0$ and $\alpha_K > 0$ such that for every $0 < \varepsilon \le \alpha_K$, for every $x \in K$ and $T \ge 1$, there exists a set $C(x, \varepsilon, T)$ satisfying $B(x, \varepsilon, T) \subset C(x, \varepsilon, T)$, such that for every $y \in B(x, \varepsilon, T)$, and every $0 < s < \Delta_K$,

$$ \varphi_{J_{C(x,\varepsilon,T)}^{\max}}(y)+s(y) \notin C(x, \varepsilon, T), \quad \varphi_{J_{C(x,\varepsilon,T)}^{\min}}(y)-s(y) \notin C(x, \varepsilon, T) \quad \text{and} \quad \text{Leb}(J_{C(x,\varepsilon,T)}(y)) \le 2\Delta_K. $$

In other words, $C(x, \varepsilon, T)$ contains portion of orbits of length at most $2\Delta_K$, after exiting $C(x, \varepsilon, T)$ an orbit remains outside $C(x, \varepsilon, T)$ for a time bounded below by $\Delta_K$ and a similar property is satisfied in the past.

*Proof.* For every $x \in K$, we can find $0 < r(x) < 1$ and $0 < \tau_0(x) < \tau_K/4$ such that $x$ admits a flow-box neighbourhood $\Omega(x)$ diffeomorphic to $B(0, r(x)) \times (-\tau_0(x), \tau_0(x))$. Let $\psi: \Omega \to B(0, r(x)) \times (-\tau_0(x), \tau_0(x))$ be the associated flow-box chart. There exists $\varepsilon_0(x) > 0$ small enough such that $B(x, \varepsilon_0(x), 1)$ is included in a flow box of half height. More precisely, for every $T \ge 1$,

$$ \psi(B(x, \varepsilon_0(x), T)) \subset \psi(B(x, \varepsilon_0(x), 1)) \subset B(0, r(x)) \times (-\tau_0(x)/2, \tau_0(x)/2). $$

Fix $0 < \varepsilon \le \varepsilon_0(x)$. It is not clear to us whether $\psi(B(x, \varepsilon, T))$ is convex (at least in the direction of the flow) or not so an orbit may exit $B(x, \varepsilon, T)$ for a very short time. To avoid these technical problems, we fill $B(x, \varepsilon, T)$ in the direction as the flow, and define $C(x, \varepsilon, T)$ as

$$ \psi^{-1}(\{(z, \tau) \in B(0, r(x)) \times (-\tau_0(x)/2, \tau_0(x)/2), \exists \tau' \in (-\tau_0(x)/2, \tau_0(x)/2), (z, \tau') \in \psi(B(x, \varepsilon, T))\}). $$

Then, by construction, $\psi(C(x, \varepsilon, T)) \subset B(0, r(x)) \times (-\tau_0(x)/2, \tau_0(x)/2)$ so that for every $0 < s < \tau_0(x)/2$ and every $y \in B(x, \varepsilon, T)$,

* $\varphi_{J_{C(x,\varepsilon,T)}^{\max}}(y)+s(y) \cap C(x, \varepsilon, T) = \emptyset$;

* $\varphi_{J_{C(x,\varepsilon,T)}^{\min}}(y)-s(y) \cap C(x, \varepsilon, T) = \emptyset$;

* $\text{Leb}(J_{C(x,\varepsilon,T)}) \le \tau_0(x)$

As $K$ is compact, $r$, $\tau_0$ and $\varepsilon_0$ can be chosen uniformly in $x$. The result follows with $\Delta_K = \tau_0/2$ and $\alpha_K = \varepsilon_0$. □

The upper bound for $m_{K_0,L}(B(x, \varepsilon, T))$ is proven in the second key lemma.
**Lemma 7.5.** Let $K_0$ be a compact set with nonempty interior as in Section 6.1, and $K \supset K_0$ be a larger compact set. There exist $\varepsilon_{K_0,K} > 0$, $\delta_{K_0,K} > 0$ and $D_{K_0,K} > 0$ such that for every $0 < \varepsilon < \varepsilon_{K_0,K}$ and every $0 < \delta < \delta_{K_0,K}$, for every $x \in K$ and for all $1 \le T < L$ such that $\varphi_T(x) \in K$, the following inequality holds

$$m_{K_0,L}(B(x, \varepsilon, T)) \le D_{K_0,K} \frac{\mathcal{N}_C(\varphi_T(x), x, \varepsilon, L-T, L-T+5\tau_{K_0}, \delta)}{L \# \mathcal{P}_{K_0}(L, L+5\tau_{K_0})} + \frac{\# \mathcal{P}_{K_0} \left( \frac{L+5\tau_{K_0}}{2} \right)}{\# \mathcal{P}_{K_0}(L, L+5\tau_{K_0})}.$$

*Proof.* The proof follows the same lines as the one of Lemma 7.3.

**Step 1.** Choice of constants. Lemma 7.4 gives us constants $\Delta_K$ and $\alpha_K > 0$ associated with $K$. Lemma 2.13 applied with $\nu = \min(\Delta_K/3, 1)$ and $\tau_1 = 5\tau_{K_0} \ge 1$ gives us $\tau_0 > 0$ and $\varepsilon_1 > 0$. Let $\varepsilon_{K_0,K} = \min(\alpha_K, \varepsilon_1)$ and $\delta_{K_0,K} = \frac{\varepsilon_1}{2}$. Fix $\varepsilon < \varepsilon_{K_0,K}$ and $\delta < \delta_{K_0,K}$. Fix $x \in K$. Fix $1 \le T \le L$ such that $\varphi_T(x) \in K$.

**Step 2.** First upper bound for $m_{K_0,L}(B(x, \varepsilon, T))$. Denote by $\mathcal{J}(L+5\tau_{K_0})$ the set of intervals included in $[0, L+5\tau_{K_0}]$. For each periodic orbit $\gamma \in \mathcal{P}_{K_0}(L, L+5\tau_{K_0})$, choose a parametrization with an origin $\gamma(0) \notin C(x, \varepsilon, T)$. Denote by $\tilde{\Theta}_{K_0,L}(x, \varepsilon, T)$ the set of pairs $(\gamma, I) \subset \mathcal{P}_{K_0}(L, L+5\tau_{K_0}) \times \mathcal{J}(L+5\tau_{K_0})$ such that for every $\gamma \in \mathcal{P}_{K_0}(L, L+5\tau_{K_0})$ (with its parametrization and associated origin), and every $s \in I$, we have $\gamma(s) \in C(x, \varepsilon, T)$, $I$ is maximal for this property and $\gamma(I) \cap B(x, \varepsilon, T) \ne \emptyset$. Note that for all $(\gamma, I) \in \tilde{\Theta}_{K_0,L}(x, \varepsilon, T)$, by Lemma 7.4, we have $\text{Leb}(I) \le 2\Delta_K$.

In the proof it will be important to focus on primitive orbits. Recall that $\mathcal{P}'_{K_0}(L, L+5\tau_{K_0})$ is the subset of primitive orbits and observe that if $\gamma$ is not primitive, then there exists a primitive periodic orbit with length at most $\ell(\gamma)/2$ with the same image. Therefore,

$$\# (\mathcal{P}_{K_0}(L, L+5\tau_{K_0}) \setminus \mathcal{P}'_{K_0}(L, L+5\tau_{K_0})) \le \#\mathcal{P}_{K_0} \left( \frac{L+5\tau_{K_0}}{2} \right).$$

We then have

$$
\begin{align*}
m_{K_0,L}(B(x, \varepsilon, T)) &= \frac{1}{\#\mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \sum_{\gamma \in \mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \mu_{\gamma}(B(x, \varepsilon, T)) \\
&\le \frac{1}{\#\mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \sum_{\gamma \in \mathcal{P}_{K_0}(L, L+5\tau_{K_0}), \gamma \text{ primitive}} \mu_{\gamma}(C(x, \varepsilon, T)) \\
&\quad + \frac{1}{\#\mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \sum_{\gamma \in \mathcal{P}_{K_0}(L, L+5\tau_{K_0}), \gamma \text{ non-primitive}} \mu_{\gamma}(B(x, \varepsilon, T)) \\
&\le \frac{1}{\#\mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \sum_{(\gamma, I) \in \tilde{\Theta}_{K_0}(L, L+5\tau_{K_0})} \frac{2\Delta_K}{L} \\
&\quad + \frac{\#\{\gamma \in \mathcal{P}_{K_0}(L, L+5\tau_{K_0}), \gamma \text{ non-primitive}\}}{\#\mathcal{P}_{K_0}(L, L+5\tau_{K_0})} \\
&\le \frac{2\Delta_K \times \#\tilde{\Theta}_{K_0}(L, L+5\tau_{K_0})}{L \#\mathcal{P}_{K_0}(L, L+5\tau_{K_0})} + \frac{\#\mathcal{P}_{K_0} \left( \frac{L+5\tau_{K_0}}{2} \right)}{\#\mathcal{P}_{K_0}(L, L+5\tau_{K_0})}
\end{align*}
$$

**Step 3.** From $\tilde{\Theta}_{K_0}(L, L+5\tau_{K_0})$ to chords. Every pair $(\gamma, I) \in \tilde{\Theta}_{K_0}(L, L+5\tau_{K_0})$ gives us a set $\{\gamma(s), s \in I\}$ of points of $\gamma$ inside $C(x, \varepsilon, T)$, with at least some $s_0 \in I$ with $\gamma(s_0) \in B(x, \varepsilon, T)$. Following $\gamma$ from $\gamma(s_0+T)$ to $\gamma(s_0+\ell(\gamma)) = \gamma(s_0)$ defines a chord with length $\ell(\gamma) - T \in [L-T, L+5\tau_{K_0}-T)$ from $B(\varphi_T(x, \varepsilon))$ to $B(x, \varepsilon)$.

Let $E$ be a $E(\varphi_T(x), x, \varepsilon, L-T, L-T+5\tau_{K_0}, \delta)$-set with maximal cardinality. In particular, $\#E = \mathcal{N}_C(\varphi_T(x), x, \varepsilon, L-T, L-T+5\tau_{K_0}, \delta)$.

For each pair $(\gamma, I) \in \tilde{\Theta}_{K_0}(L, L+5\tau_{K_0})$, choose a point $z \in E$, such that $\gamma(s_0+T) \in B(z, \delta, L-T)$. This gives us a map $\theta: \tilde{\Theta}_{K_0}(L, L+5\tau_{K_0}) \to E$.

**Step 4.** Control the (lack of) injectivity of $\theta$. Let $(\gamma_1, I_1)$ and $(\gamma_2, I_2)$ be two pairs that lead to the same point $z \in E$. In particular, there exist $s_1 \in I_1, s_2 \in I_2$ such that $\gamma_1(s_1), \gamma_2(s_2) \in B(x, \varepsilon, T)$,
so, for every $0 \le s \le T$,

$$d(\gamma_1(s_1 + s), \varphi_s(x)) \le \varepsilon \quad \text{and} \quad d(\gamma_2(s_2 + s), \varphi_s(x)) \le \varepsilon.$$

Moreover, as $\theta(\gamma_1, I_1) = \theta(\gamma_2, I_2) = z$, for every $0 \le s \le L - T$, we have

$$d(\gamma_1(s_1 + T + s), \gamma_2(s_2 + T + s)) \le d(\gamma_1(s_1 + T + s), \varphi_s(z)) + d(\varphi_s(z), \gamma_2(s_2 + s + T)) \le 2\delta.$$

Therefore, for all $0 \le s \le L$, we have

$$d(\gamma_1(s_1 + s), \gamma_2(s_2 + s)) \le \varepsilon_1.$$

If $0 \le \ell(\gamma_1) - \ell(\gamma_2) \le \tau_0$, using Lemma 2.13 (see Step 1) and the fact that $\gamma_1$ and $\gamma_2$ are primitive, we deduce that $\gamma_1 = \gamma_2$ and there exists $u \in [-\nu, \nu]$ such that $s_2 = s_1 + u$. By Lemma 7.4, we deduce that $I_1 = I_2$ (otherwise, we would have $|s_1 - s_2| = |u| \ge \Delta_k$, contradicting $|u| \le \nu = \min(\frac{\Delta_k}{3}, 1)$).

Cutting the interval $[L, L + 5\tau_{K_0}]$ into intervals of length $\tau_0$, we deduce that the number of elements $(\gamma, I) \in \tilde{\Theta}_{K_0}(L, L + 5\tau_{K_0})$ leading to the same chord is bounded from above by $5\tau_{K_0}/\tau_0$.

The result of the lemma follows with $D_{K_0,K} = 10\Delta_K \tau_{K_0}/\tau_0$. $\square$

## 7.2 Estimation of the measure of dynamical balls

In this section, we gather all the inequalities proven in Lemma 7.3, Lemma 7.5, Proposition 6.4 and Proposition 6.5 to obtain the following strong inequalities.

Let $K_0$ be a compact set as in Section 6.1. Let $m_\infty$ be any accumulation point of the family $(m_{K_0,L})_L$ and $m_{\max}$ be the probability measure obtained after renormalizing $m_\infty$. In particular, $m_\infty(K_0) > 0$.

Choose $K \supset \tilde{K} \supset K_0$ such that $m_\infty(\tilde{K}) > \frac{3}{4}m_\infty(M)$.

Choose an increasing sequence $(L_k)_k$ such that $L_k \to +\infty$ as $k \to +\infty$ and $m_{K_0,L_k} \xrightarrow{\ast} m_\infty$. Fix some point $y_0 \in K_0$.

**Theorem 7.6.** There exist $\delta_{\text{end}} > 0$, such that for every $0 < \delta < \delta_{\text{end}}$, there exists $\varepsilon_{\text{end},\delta} > 0$ such that for every $0 < \varepsilon < \varepsilon_{\text{end},\delta}$ there exists $\eta_{\text{end},\delta,\varepsilon}$ such that for every $0 < \eta < \eta_{\text{end},\delta,\varepsilon}$, there exist positive constants $S^-, S^+, D^-, D^+ > 0$, and $T_{\text{end}} > 0$ such that the following holds. For every $x \in K$ and $T > T_{\text{end}}$ such that $\varphi_T(x) \in K$, we have

$$\frac{D^{-}}{\mathcal{N}_{C}(x, y_{0}, \eta, T + S^{-}, T + S^{-} + 5\tau_{K_{0}}, \delta)} \leq m_{\infty}(B(x, \varepsilon, T)) \leq \frac{D^{+}}{\mathcal{N}_{C}(x, y_{0}, \varepsilon, T - S^{+}, T - S^{+} + 5\tau_{K_{0}}, \delta)}. \qquad (30)$$

Before proving it, let us emphasize the strength of this statement. Usually, in ergodic theory, invariant measures satisfy almost sure properties. The above inequalities hold for every $x \in K$, and are therefore more geometric than ergodic. This strong uniform property will allow us to conclude that $m_{\max}$ is a measure of maximal entropy.

*Proof.* The proof follows easily from the preceding work, and in particular from Propositions 6.4 and 6.5, and from Lemmas 7.3 and 7.5, as soon as parameters are carefully chosen. We start with this choice.

**Step 1.** Choice of constants. Set $\delta_{\text{end}} = \min(\frac{\varepsilon_1}{4}, \delta_{K_0,K}, 1)$ where $\varepsilon_1$ is given by Proposition 6.5 and $\delta_{K_0,K}$ by Lemma 7.5. Choose an arbitrary $0 < \delta < \delta_{\text{end}}$.

Set $\varepsilon_{\text{end},\delta} = \min(\varepsilon_{K_0,K}, \varepsilon_{K,\delta})$, where $\varepsilon_{K_0,K}$ is given by Lemma 7.5 and $\varepsilon_{K,\delta}$ by Lemma 7.3 applied with parameter $\delta$. Fix $0 < \varepsilon < \varepsilon_{\text{end},\delta}$.

Lemma 7.3 applied with $\varepsilon/2$ gives constants $\sigma_{K,\delta,\varepsilon/2}$ and $\eta_{K,\delta,\varepsilon/2}$. Set $\eta_{\text{end},\delta,\varepsilon} = \min(\delta, \eta_{K,\delta,\varepsilon/2}, 1)$. Choose $0 < \eta < \eta_{\text{end},\delta,\varepsilon}$.

Propositions 6.4 (with parameters $\varepsilon$ and $\delta$) and 6.5 (with parameters $\eta$ and $\delta$) give constants $S_1, D_1, S_2, D_2 > 0$. Set $T_{\text{end}} = \max(T_{\text{min}} + S_1, 5\tau_{K_0}, T'_{\text{min}})$, where $T_{\text{min}}$ is given by Proposition 6.4 and $T'_{\text{min}}$ by Lemma 7.3. Set $S^+ = S_1$ and $S^- = 2S_2 + \sigma_{K,\delta,\varepsilon/2}$, $D^+ = D_{K_0,K}D_1$ and $D^- = \frac{\varepsilon}{8bD_2}$.

Let $T \ge T_{\text{end}}$. Let $x, y_0 \in K$.
**Step 2.** Upper bound. By Lemma 7.5 and Proposition 6.4 applied with $T_0 = L_k - T$, $T_1 = T - S_1$ and $S = S_1$, as soon as $L_k > T + T_{min}$, we have

$$m_{K_0, L_k}(B(x, \varepsilon, T)) \le D_{K_0, K} \frac{\mathcal{N}_C(\varphi_T(x), x, \varepsilon, L_k - T, L_k - T + 5\tau_{K_0}, \delta)}{L_k \# \mathcal{P}_{K_0}(L_k, L_k + 5\tau_{K_0})} + \frac{\# \mathcal{P}_{K_0} \left( \frac{L_k + 5\tau_{K_0}}{2} \right)}{\# \mathcal{P}_{K_0}(L_k, L_k + 5\tau_{K_0})} \\ \le \frac{D_{K_0, K} D_1}{\mathcal{N}_C(x, y_0, T - S_1, T - S_1 + 5\tau_{K_0})} + \frac{\# \mathcal{P}_{K_0} \left( \frac{L_k + 5\tau_{K_0}}{2} \right)}{\# \mathcal{P}_{K_0}(L_k, L_k + 5\tau_{K_0})}.$$

Using that $B(x, \varepsilon, T)$ is open and from Corollary 3.11, we get

$$m_{\infty}(B(x, \varepsilon, T)) \le \liminf_{k \to +\infty} m_{K_0, L_k}(B(x, \varepsilon, T)) \le \frac{D_{K_0, K} D_1}{\mathcal{N}_C(x, y_0, T - S_1, T - S_1 + 5\tau_{K_0})}.$$

The upper bound follows with $D^+ = D_{K_0, K} D_1$.

**Step 3.** Lower bound. Apply Lemma 7.3 with parameter $\varepsilon/2$ and Proposition 6.5 with $S = S_2$ and $T = T + S_2 + \sigma_{K, \delta, \varepsilon/2} > 5\tau_{K_0}$, and $x_1 = \varphi_T(x), y_1 = x_0 = x$. Then, we get, for $k$ big enough,

$$m_{K_0, L_k}(B(x, \varepsilon/2, T)) \ge \frac{\varepsilon}{8b} \times \frac{\mathcal{N}_C(\varphi_T(x), x, \eta, L_k - T - \sigma_{K, \delta, \varepsilon/2}, L_k - T - \sigma_{K, \delta, \varepsilon/2} + 5\tau_{K_0}, \delta)}{L_k \# \mathcal{P}_{K_0}(L_k, L_k + 5\tau_{K_0})} \\ \ge \frac{\varepsilon}{8b} \times \frac{1}{D_2} \times \frac{1}{\mathcal{N}_C(x, y_0, \eta, T + 2S_2 + \sigma_{K, \delta, \varepsilon/2}, T + 2S_2 + \sigma_{K, \delta, \varepsilon/2} + 5\tau_{K_0}, \delta)}.$$

As $\limsup_{L_k \to \infty} m_{K_0, L_k}(B(x, \varepsilon/2, T)) \le m_{\infty}(\overline{B(x, \varepsilon/2, T)}) \le m_{\infty}(B(x, \varepsilon, T))$, the lower bound follows with $D^- = \frac{\varepsilon}{8bD_2}$ and $S^- = 2S_2 + \sigma_{K, \delta, \varepsilon/2}$. $\square$

## 7.3 Computation of entropies of $m_{\max}$

**Theorem 7.7.** Let $\varphi$ be a H-flow on $M$ such that $h_{\text{Gur}}^{\infty}(\varphi) < h_{\text{Gur}}(\varphi)$. Let $m_{\max}$ be the probability measure obtained after renormalizing an accumulation point of the family of measures distributed on periodic orbits of increasing length, see Section 6.1. Then

$$\underline{h}_{\text{BK}}(m_{\max}) = \bar{h}_{\text{BK}}(m_{\max}) = h_{\text{KS}}(m_{\max}) = h_{\text{Kat}}(m_{\max}) = h_{\text{Gur}}(\varphi).$$

*Proof.* As at the beginning of Section 7.2, choose $K_0$, and $L_n \to \infty$ such that $m_{K_0, L_n} \stackrel{*}{\to} m_{\infty}$, and $K \supset \check{K} \supset K_0$ such that $m_{\infty}(\check{K}) > \frac{3}{4}m_{\infty}(M)$. Fix $\delta < \min(\alpha_0/2, \delta_{\text{end}})$ where $\delta_{\text{end}}$ comes from Theorem 7.6 and $\alpha_0$ from Proposition 5.12. Fix $\varepsilon < \min(\varepsilon_{\text{end}, \delta}, \alpha_0/4)$, where $\varepsilon_{\text{end}, \delta}$ comes from Theorem 7.6 with parameter $\delta$, such that $\varepsilon$ is small enough compatibly with Proposition 4.1. Let $x \in K$.

Theorems 7.6 and Proposition 5.12 give us

$$\limsup_{T \to \infty} \frac{1}{T} \log m_{\infty}(B(x, T, \varepsilon)) = \liminf_{\varphi_T(x) \in K} \frac{1}{T} \log m_{\infty}(B(x, T, \varepsilon)) = h_{\text{Gur}}(\varphi).$$

Therefore

$$\underline{h}_{BK}(m_{\max}) = \bar{h}_{BK}(m_{\max}) = h_{\text{Gur}}(\varphi).$$

From Proposition 4.1 we also have

$$\int_K \limsup_{\varphi_T(x) \in K} \frac{1}{T} \log(m_{\infty}(B(x, \varepsilon, T))) dm_{\max} \le h_{\text{KS}}(m_{\max}).$$

Thus we obtain the lower bound $h_{\text{KS}}(m_{\max}) \ge h_{\text{Gur}}(\varphi)$.

Corollary 4.4 gives the inequality $h_{\text{KS}}(m_{\max}) \le h_{\text{Gur}}(\varphi)$ so that $h_{\text{KS}}(m_{\max}) = h_{\text{Gur}}(\varphi)$. We know by Theorem 4.3 that $h_{\text{Kat}}(m_{\max}) \le h_{\text{Gur}}(\varphi)$.

It only remains to prove the inequality $h_{\text{Kat}}(m_{\max}) \ge h_{\text{Gur}}(\varphi)$. By definition of Katok entropy, for every $0 < \nu < 1$, there exist $\alpha_{\nu} > 0$, $\varepsilon_{\nu} > 0$ and $T_{\nu} > 0$ such that, for every $0 < \alpha < \alpha_{\nu}$, $0 < \varepsilon < \varepsilon_{\nu}$
and $T \ge T_\nu$, the minimal cardinality $M(T, \varepsilon, \alpha, m_{\max})$ of a set of $(\varepsilon, T)$-dynamical balls covering a set of measure at least $\alpha$ satisfies

$$M(T, \varepsilon, \alpha, m_{\max}) \le e^{(h_{\mathrm{Kat}}(m_{\max})+\nu)T}.$$

Even if it means to increase $K$, we can assume that $m_{\max}(K) \ge 1 - \frac{\alpha}{4}$. By invariance of $m_{\max}$, we get $m_{\max}(K \cap \varphi_{-T}(K)) \ge 1 - \frac{\alpha}{2}$. Without loss of generality, we can assume that $T$ is large enough and $\varepsilon > 0$ small enough so that Theorem 7.6 holds with parameters $K$, $\delta$ (where $\delta$ is some small enough constant), $2\varepsilon$ and $T$. We may also assume that Proposition 5.12 holds with parameters $K, C = 5\tau_{K_0}, \delta$ and $\eta = 2\varepsilon$.

Since $m_{\max}(K \cap \varphi_{-T}(K)) \ge 1 - \frac{\alpha}{2}$, we can use Lemma 4.9 to obtain a separating $(T, 2\varepsilon, \frac{\alpha}{2}, m_{\max})$-spanning set $E' \subset K \cap \varphi_{-T}(K)$ with $\#E' \le M(T, \varepsilon, \alpha, m_{\max})$. Therefore

$$\begin{aligned} \frac{\alpha}{2} \le m_{\max}(\cup_{x \in E'} B(x, 2\varepsilon, T)) &\le \sum_{x \in E'} m_{\max}(B(x, 2\varepsilon, T)) \\ &\le M(T, \varepsilon, \alpha, m_{\max}) \times \max_{x \in E'} m_{\max}(B(x, 2\varepsilon, T)) \\ &\le e^{(h_{\mathrm{Kat}}(m_{\max})+\nu)T} \times \max_{x \in E'} m_{\max}(B(x, 2\varepsilon, T)). \end{aligned}$$

Using Theorem 7.6 we get

$$m_{\max}(B(x, 2\varepsilon, T)) = \frac{1}{m_{\infty}(M)} m_{\infty}(B(x, 2\varepsilon, T)) \le \frac{1}{m_{\infty}(M)} \frac{D^+}{\mathcal{N}_C(x, y_0, 2\varepsilon, T - S^+, T - S^+ + 5\tau_{K_0}, \delta)},$$

so that

$$\frac{\alpha}{2D^+} \times m_{\infty}(M) \times \mathcal{N}_C(x, y_0, 2\varepsilon, T - S^+, T - S^+ + 5\tau_{K_0}, \delta) \le e^{(h_{\mathrm{Kat}}(m_{\max})+\nu)T}.$$

Thus,

$$\frac{1}{T} \log \left( \frac{\alpha}{2D^+} \times m_{\infty}(M) \right) + \frac{1}{T} \log \left( \mathcal{N}_C(x, y_0, 2\varepsilon, T - S^+, T - S^+ + 5\tau_{K_0}, \delta) \right) \le h_{\mathrm{Kat}}(m_{\max}) + \nu.$$

Taking the limit superior when $T \to +\infty$ of the above inequality, by Proposition 5.12, we obtain $h_{\mathrm{Gur}}(\varphi) \le h_{\mathrm{Kat}}(m_{\max}) + \nu$. As $\nu$ can be chose arbitrarily small, we have $h_{\mathrm{Gur}}(\varphi) \le h_{\mathrm{Kat}}(m_{\max})$. This concludes the proof of the theorem. $\square$

# 8 Notations

* $B(x, \varepsilon, T)$ - p. 15

* $C(x, y, \eta, T^-, T^+)$ - p. 31

* $C^{K^c}(x, y, \eta, T^-, T^+)$ - p. 39

* $E(x, y, \eta, T^-, T^+, \delta)$ - p. 31

* $E^{K^c}(x, y, \eta, T^-, T^+, \delta)$-set - p. 39

* $E_K(x, y, \eta, T^-, T^+, \delta)$-set - p. 37

* $h_C(x, y, \eta, \delta)$ - p. 37

* $h_C(\varphi)$ - p. 39

* $h_C^{K^c}(x, y, \eta, \delta)$ - p. 40

* $h_C^{K^c}(\eta, \delta)$ - p. 40

* $h_C^{K^c}(\varphi)$ - p. 40

* $h_C^\infty(\varphi)$ - p. 41

* $\underline{h}_{\mathrm{BK}}(\mu)$ - p. 16

* $\bar{h}_{\mathrm{BK}}(\mu)$ - p. 16

* $h_{\mathrm{Gur}}(\varphi)$ - p. 17

* $h_{\mathrm{Kat}}(\mu)$ - p. 16

* $h_{\mathrm{KS}}(\mu)$ - p. 15

* $h_{\mathrm{var}}(\varphi)$ - p. 15

* $h_{\mathrm{Gur}}^{K, \alpha}$ - p. 22

* $h_{\mathrm{Gur}}^\infty$ - p. 22

* $h_{\mathrm{var}}^\infty$ - p. 23

* $K_{-\eta}$ - p. 39

* $\ell$ - p. 7
• $m_{K,L}$ - p. 54

• $m_{\max}$ - p. 55

• $m_{\infty}$ - p. 55

• $M(T, \varepsilon, \alpha, \mu)$ - p. 16

• $M'(T, \varepsilon, \alpha, \mu)$ - p. 25

• $M_{\varphi}$ - p. 15

• $M_{\varphi}^{\text{erg}}$ - p. 15

• $N_C(x, y, \eta, T^{-}, T^{+}, \delta)$ - p. 31

• $N_C^{K^c}(x, y, \eta, T^{-}, T^{+}, \delta)$ - p. 39

• $P_K(T_0)$ - p. 16

• $P_K(T_0, T_1)$ - p. 16

• $P_K^{\alpha}(L, L + C)$ - p. 22

• $P(K_1, K_2, \alpha, T, T + C)$ - p. 46

• $\tau_K$ - p. 11

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