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I'm looking for theorems that can be used to show that a topological partition for a given expanding map is Markov. Here are the relevant definitions:

  1. Let $\phi\colon\mathbb{R}^m\to\mathbb{R}^m$ be a $C^1$ map, let $J\subseteq\mathbb{R}^m$ be a compact set, and suppose that $\phi(J) = J$.

  2. We say that $\phi$ is expanding on $J$ if there exists an $n\in\mathbb{N}$ so that $\|D[\phi^n]_x(v)\|>\|v\|$ for all $x\in J$ and all nonzero $v\in\mathbb{R}^n$.

  3. A topological partition of $J$ is a finite collection $U_1,\ldots,U_k$ of (relatively) open, disjoint subsets of $J$ whose closures cover $J$.

  4. A topological partition $U_1,\ldots,U_k$ of $J$ is called a Markov partition if it satisfies the following conditions:

    1. For each $i$ and $j$, either $\phi(U_i)\cap U_j = \emptyset$ or $U_j\subseteq \phi(U_i)$
    2. .
    3. For every sequence $i_0,i_1,i_2,\ldots$, the intersection $\bigcap_{k=0}^\infty \phi^{-k}\bigl(\\,\overline{U_{i_k}}\\,\bigr)$ contains at most one point.

    (These conditions guarantee that $\phi$ is semi-conjugate to a subshift of finite type.)

My question is, how can we tell that a given topological partition $U_1,\ldots,U_k$ is Markov? For example, suppose that a partition satisfies condition (1) above for a Markov partition, and that $\phi$ is expanding on $J$ and one-to-one on each $\overline{U_i}$. Does it follow that $U_1,\ldots,U_k$ satisfies condition (2) above? If not, what extra hypotheses are required?

If it helps, the $J$ that I am interested in is the connected Julia set for a hyperbolic rational map on the Riemann sphere, and each $U_i$ is connected. I have a specific Markov partition that I want to prove is Markov, and I would prefer to simply cite some theorem.

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2 Answers

If you can prove something like: $d(\phi^n(x),\phi^n(y))>(1+\delta)d(x,y)$ for all $x$ and $y$ such that $d(x,y)<\epsilon$, then it would suffice to show that there exists a $k$ such that for all sequences $i_0,i_1,\ldots,i_{nk-1}$, $\bigcap_{j=0}^{nk-1}\phi^{-j}\overline{U_{i_j}}$ has diameter at most $\epsilon$.

If this latter condition is not satisfied, then condition 2 will not be satisfied anyway, so this is a necessary condition.

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If you want to give a Markov coding for an expanding dynamical system you usually prove that the local inverse maps are contractions on the elements of your partition. Condition 2. than follows by Banach's fix point theorem.

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