There are only finitely many classes of Markov partitions for pseudo-Anosov diffeomorphisms of surfaces (Q736239)

The author considers a given pseudo-Anosov diffeomorphism \(T\) of a surface \(M\). A \textit{pre-Markov partition} is a partition \[ \Xi =\left( X_1,\dots ,X_m\right) \] fulfilling the following properties: 1) Each \(X_j\) is a parallelogram; i.e., there exists a continuous mapping \(h_j :\left[ 0;1\right] ^2\to X_j\) whose restriction to \(\left( 0;1\right) ^2\) is a homeomorphism, \(h_j\left(\left[ 0;1\right]\times\left\{ y\right\}\right)\) is a segment of a leaf of the stable foliation \(\mathcal F ^s\) for \(T\) for each \(y\in\left( 0;1\right)\) and \(h_j\left(\left\{ x\right\}\times\left[ 0;1\right]\right)\), \(x\in\left( 0;1\right)\), is a segment of a leaf of the unstable foliation \(\mathcal F ^u\). The set \(h_j\left(\left\{ 0,1\right\}\times\left[ 0;1\right]\right)\) will be called the unstable boundary of \(X_j\) and denoted by \(\partial ^u X_j\). Likewise, the set \(h_j\left(\left[ 0;1\right]\times\left\{ 0,1\right\}\right)\) will be called the stable boundary \(\partial ^s X_j\). We also set \[ \partial ^{u,s}\Xi =\bigcup\partial ^{u,s} X_j\;\;,\;\;\partial\Xi =\partial ^s\Xi\cup\partial ^u\Xi\;. \] 2) \(T\left(\partial ^u\Xi\right)\supset\partial ^u \Xi\), \(T\left(\partial ^s\Xi\right)\subset\partial ^s\Xi\). Partitions \(\Xi ^{(1)}\) and \(\Xi ^{(2)}\) are said to be \textit{equivalent} if \(\Xi ^{(2)} =T^k\left(\Xi^{(1)}\right)\) for some integer \(k\). The only theorem of the present paper is: There exist only finitely many equivalence classes of pre-Markov partitions with a given number of partition elements.