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Let $M$ be $S^1 \times [-1,1]$, $f$ a baker map on $M$ and for $p, q \in M$ consider $W^s_p$ the stable manifold in $p$ (i.e. the set of points whose forward orbit tend to the forward orbit of $p$) and $W^u_q$ the unstable manifold in $q$ (i.e. the set of points whose backward orbit tends to backwards orbit of $q$). Let us assume, we have transversality. Question:

Let $x\in W^u_p$ and $y\in W^u_q$. Show that there exists a point $t$ so that $t\in W^u_x \cap W^u_y$, if $x$ and $y$ be small close to each other. I want to say that if two point be close to each other their unstable manifold will intersect each other at a point.

Under which assumption we can prove it the intersection point is unique?

When we draw baker map we easily see they intersect a point but i do not know how can i prove it?

Thanks in advance.

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    $\begingroup$ It's not true. In fact if we assume that $f$ is Anosov then locally the stable manifolds form a decomposition, hence the only way that that $W^u_x$ and $W^u_y$ can have nonempty intersection is if they are equal. Are you sure you wrote the question correctly? $\endgroup$
    – Lee Mosher
    Commented May 18, 2019 at 0:43
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    $\begingroup$ Is the result you want that (under some conditions) the stable manifold of $x$ meets the unstable manifold of $y$? $\endgroup$
    – Anthony Quas
    Commented May 18, 2019 at 1:16
  • $\begingroup$ @LeeMosher : Thanks for your comment. I forgot to write map is Baker. When we draw picture we will see their unstable manifold intersect each other but I can not prove it $\endgroup$
    – Adam
    Commented May 18, 2019 at 15:30
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    $\begingroup$ @AnthonyQuas : Thanks for your comment. No, I know when we have local product structure stable and unstable manifold intersect each other . But as I said I want to show that unstable manifolds intersect each other in Baker map. We see they intersect each other when we draw it but I do not know how I can prove it. $\endgroup$
    – Adam
    Commented May 18, 2019 at 15:32
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    $\begingroup$ Perhaps you can provide a definition or a link to a "Baker map". I know what the Baker's map is, but presumably that is not what you mean because its stable and unstable manifolds do indeed have a local product structure. $\endgroup$
    – Lee Mosher
    Commented May 18, 2019 at 15:40

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