Question:
I have proved that if $\Lambda$ is a hyperbolic set that has the local product property, i.e., there exists $\epsilon > 0$ and $\delta > 0$ such that if $d(x,y) < \delta$ then $[x,y] = W_{\varepsilon}^s (x) \cap W_{\varepsilon}^u (y)$ has only one point contained in $\Lambda$ then $$W_\varepsilon^s (\Lambda) \subseteq \bigcup_{x \in \Lambda} W_r^s(x) \tag {1}$$ for some $r \leq 0$.
It remains to show if $\Lambda$ does not have the local product property then $(1)$ does not necessarily hold.
Thoughts: So far the only example I have that does not have the local product property is $\Lambda = \mathcal O(x) \cup \{p\}$, where $p$ is a periodic point and $x$ a homoclinic point. Unfortunately, this example satisfies $(1)$.
Any leads on this? I really have no clues here.
The example comes from symbolic dynamics. Consider $\Lambda$ as the set of sequences $(x_i)_{i \in \mathbb N}$ in $\Sigma_2$ with blocks of zero with even size. Now consider
$$x_0 = (\ldots 111.01000100000100000001...)$$ where we have 0 followed by 1 and 3 zeros, then 1 and 5 zeros, and so on, always with an odd number of zeros.
Then for $\varepsilon > 0$ given, take $N \in \mathbb N$ suh that $\frac{1}{2^{N-2}} < \varepsilon$. Then there exists $x \in \Lambda$ that coincides with $x_0$ (centered at zero) around its $N-2$ coordinates and
$$\begin{align} d(\sigma^n (x), \sigma^n (x_0)) &= \sum_{n \geq N+1} \frac{|\sigma^n (x) - \sigma^n (x_0)|}{2^{|n|}}\\&\leq \sum_{n\geq N+1} \frac{1}{2^n} = \frac{1}{2^{N-2}} < \varepsilon\end{align}$$ so $X_0 \in W^s_\varepsilon (\Lambda)$.
On the other hand, there is no $p \in \Lambda$ such that $x \in W^s (p)$. If this were the case, $x_0$ and $p$ would coincide at a certain moment on (for every $\delta > 0$) and this cannot happen.