Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.

There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the diffeomorphism $f$ and where strong expansion of the unstable bundle $E^{u}$, strong contraction $E^{ss}$ and weak contraction $E^{s}$.

Special case, you can consider $M=S^{1}\times \mathbb{D}$ so f is solenoid map such that $$(x,y,z)\rightarrow (mx mod1, \lambda y +u(x), \mu z+v(x))$$ where $m=2$ and $\mu<\lambda<\frac{1}{2}$. $\Lambda$ is attractor.

If A,B are two nearby embedded disks transverse to $W^{u}$ then there is a $\mathit{holonomy}$ map defined on subset of $A \cap \Lambda $ by $p\rightarrow W^{u}(p)\cap B$ whenever this make sense. In other words, From A to B along unstable leaves. The holonomies are always continuous and in fact Holder continuous.There woild be no need these holonomies are always Lipschitz continuous.

**Question**
Why does Lipschitz property holonomies fails, when stable leaves $W^s(x)$ inside the
leaves $W^{ss}(x)?$

In fact,The question is paragraph of a paper(Dimension and product structure of hyperbolic measures)by Barreira,Pesin and Schmeling, unfortunately they did not explain at all.They used this result in another paper(Dimension product structure of hyperbolic sets)by Hasselbalt and Schmeling.They did not explain,again.