# How do we recognize a Markov partition?

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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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 $\epsilon$.