Let $\Lambda$ be Axiom A for $C^{1+\gamma}$ $f$. I am reading this paper. I have a problem to undestand holonomies. The holonomy mapping $$ h: W_{loc}^{s} (x) \cap\Lambda \rightarrow W_{loc}^{s} (y) \cap \Lambda$$ is defined by $h(x)=W_{loc}^{s} (x) \cap W^{u}(x)$. That means we move long unstable manifold from $W_{loc}^{s} (x)$ to $W_{loc}^{s} (y)$. I know it should be a unique point the intersection on the right hand side but i do not know why. Could one explain to me ?
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$\begingroup$ I guess you mean $h(z) = W^s_{loc}(y) \cap W^u_{loc}(z)$ for $z \in W^s_{loc}(x)$. This point exists and is unique (when one defines the '$loc$' correctly) by the local product structure property. See for instance Shub's book on hyperbolic dynamics (Global stability of dynamical systems, I think). $\endgroup$– rpotrieCommented Oct 2, 2019 at 15:28
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$\begingroup$ @rpotrie : Yes, Thank you for your anwser. $\endgroup$– AdamCommented Oct 9, 2019 at 12:29
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