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Let $I=[0,1]$ and $\alpha \in ( 0,1)$. Define the Tent map, $T: I \rightarrow I$ by

$T(x)= x/\alpha$ for $x \in [0,\alpha]$ and $(1-x)/(1-\alpha)$ for $x \in [\alpha,1]$

Find the measure theoretic entropy.

For the case $\alpha =1/2$, I can calculate the Entropy. However, for the general case I am not sure how to do it. Since as soon as I start calculating $T^{-i}P$ for a given partition $P$ the expressions become very complicated.

I am guessing that I should move to a representation in terms of shift dynamics but I am not sure how. Could anybody give me a hint?

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    You work with lebesgue measure, I guess. Consider the generating partition $\{[0, \alpha),[\alpha,1)\}$, or (this is roughly equivalent) prove that your system is isomorphic to a bernoulli shift on two symbols.2012-01-17
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    Thank you for your reply. But I had already tried this. The problem that I had was that I couldn't construct a conjugacy between the two. Since the conjugacy in terms of a binary expantion for the case $\alpha=1/2$ didn't work anymore. Could I have another hint?2012-01-17

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