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For $T : (x,y) \mapsto (x+y,x+a)$: First, notice that $T=A+\varphi$ where $A= \left( \begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}\right)$ and $\varphi \equiv (0,a)$. Because $A$ is hyperbolic and $\varphi$ is $0$-Lipschitz, $T$ and $A$ are in fact conjugated. Therefore, $$h(T)=h(A)= \ln \left( \frac{1+ \sqrt{5}}{2} \right).$$ For $T : (x,y) ... 1 The "entropy" is a measure of the amount of information in a message (the name comes from many of the formulas having an uncanny resemblance to the ones in statistical mechanics for entropy). And customarily "information contents" is measured in bits (2 options), thus the "power of 2" and "logarithm of base 2" which show up all over the place. If martians ... 1 From the comments above: In binary, you only have two bits; thus encoding a character which can take 26 values requires$\lceil\log_2 26\rceil$bits, since the number of possible different elements encoded using$b$bits is$2^b$, and you need this to be at least$26$to cover all possible distinct 26 characters. This explains why the$\log_2\$ shows up ...