TITLE: Entropy of the induced transformation
QUESTION [5 upvotes]: I need help with this problem:
Let $(X,\mathcal{B},\mu,T)$ be a ergodic dynamical system in the probability space $(X,\mathcal{B},\mu)$.
Let $A \in \mathcal{B}$ with $\mu(A)>0$. We define the first return of $x\in A$ to the set $A$  by 
$$r_A(x) = \inf \{n \geq 1 \colon T^n(x) \in A\}$$
And then we can define the induced transformation in $A$ by $$T_A : A \to A \qquad T_A(x)= T^{r_A(x)}(x)$$
Now we can define the dynamical system $(A,\mathcal{B}\cap A, \mu_A = \mu( \ \cdot \ | A),T_A)$.
This construction is made in complete detail in "Ergodic Theory: with a view towards number theory" page 61.
Prove that $$h_{\mu_A}(T_A)=\frac{h_\mu(T)}{\mu(A)}$$
Any help will be appreciated! (:

REPLY [3 votes]: This is not entirely trivial. The formula was found by Abramov in 1959.
A rather simple proof (using some clever reductions) can be found in Karl Petersen's book "Ergodic theory", Cambridge University Press, 1983, Section 6.1.C, on pages 257-259; Google books link: http://books.google.com/books?id=MiyJGqFCbEMC&pg=PA257