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## The shape of Bowen balls

These balls appear in the definition of topological entropy given by Rufus Bowen.

Let $X$ be a compact metric space, $f:X\to X$ be a homeomorphism. For each $n\ge1$ the Bowen ball $B(x,\epsilon,n)$ is given by

$B(x,\epsilon,n)=\cap_{0\le k\le n-1}f^{-k}B(f^kx,\epsilon)=(y\in X:d_n(x,y)<\epsilon),$

where $d_n(x,y)=\max_{0\le k\le n-1} d(f^kx,f^ky)$.

These new metrics $d_n$ are equivalent to $d$. So they induce the same topology on $X$. In particular for each fixed $n\ge1$, there exists $\epsilon(n)$ small enough such that all Bowen balls $B(x,\epsilon,n)$ are really 'balls'.

I think, in general, the shapes of these balls could be quite wild: the scale $\epsilon(n)$ does depend on $n$.

But in differential dynamical systems (e.g. $f\in\mathrm{Diff}(M)$), it seems that these balls have quite ideal shape: the scale $\epsilon(n)$ can be chosen uniformly, such balls are simply connected, even sort of convex.

If the map $f$ is Anosov, it is OK since we have transversal stable ans unstable foliation, Markov partitions, etc. But for general maps?

So my question is, is this really the case, that these balls are ideal balls? Thanks!

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I'm not sure that I understand the question. What do you mean by "ideal balls"? – Ian Morris Jan 20 2011 at 11:21
Sorry about this. What I want to know is whether the scale $\epsilon(n)$ can be chosen uniformly, and whether these balls are simply connected or when they could be convex with respect to the original Riemannian metric on $M$. – Pengfei Jan 20 2011 at 13:35