# A argument related measurable partitions in dynamic system

$X$ is a compact metric space, and $T:X\rightarrow X$ be a continuous map, which is finite to one. Denoted by$X_{0}$ the set of all points $x\in X$, such that for all sufficiently small $\epsilon>0$, the set $TB(x,\epsilon)$ is not open. A point $x\in X$ is called critical, if there does not exist an open neighborhood of $x$, which is a special set ( special set $A$ means that $T:A\rightarrow T(A)$ is invertible. The union of $X_{0}$ and critical points is called singular points for $T$.

I was confused by the following claim:

If there are only finite may critical points for dynamics $T$, and $m$ is Borel measure, then there exists finite measurable partitions of $X(\mod m)$ into special sets on which $T$ acts as a measurable isomorphism. This is claimed in paper of on the existence of conformal measures without sufficient details.

Any reference and comments will be appreciated.

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I don't see any reason for this to be true if the measure $m$ is a Dirac measure supported on the critical points. So probably, you forgot an assumption about the measure. –  Barbara Schapira Dec 1 '13 at 14:17