Topological Entropy of $(x,y)\mapsto(x+y,x+a)$

Question

Let $a\in \mathbb T^1$, how can I calculate the topological entropies of the maps $T_1:(x,y)\mapsto(x+y,x+a)$ and $T_2 : (x,y) \mapsto (x+y,y+a)$ defined on $\mathbb T^2$. Here $\mathbb T^n$ is the torus of dimension $n$.

Answer 1

• 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 $T : (x,y) \mapsto (x+y,y+a)$:

First, a general remark. If $M$ is a compact $m$-dimensional manifold and $f : M \to M$ is $p$-Lipschitz, then $$h(f) \leq \max(0,m \ln(p)).$$ When $p\leq 1$, the fact $h(f)=0$ is usual; otherwise, if $x,y \in M$ are $(n,\epsilon)$-separated then $$\epsilon \leq d(T^kx,T^ky) \leq p^kd(x,y)$$ for some $k \leq n$. Therefore, if $K$ denotes the volume of $M$, we have (for small enough $\epsilon$): $$s(n,\epsilon) \leq \frac{K}{(\epsilon/p^n)^m}= \frac{p^{mn} \cdot K}{\epsilon^m}$$ hence $$h(f)= \lim\limits_{\epsilon \to 0} \limsup\limits_{n \to + \infty} \frac{1}{n} \ln(s(n,\epsilon)) \leq m \ln(p).$$

Here, $T^n : x \mapsto A^nx + B$ with some $B \in \mathbb{R}^2$ that does not depend on $x$ and $A= \left( \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} \right)$, so $T^n$ is $\|A^n\|$-Lipschitz (according to mean value theorem). Therefore, $$nh(T)=h(T^n) \leq 2 \ln(\|A^n\|)$$ hence $$h(T) \leq \lim\limits_{n \to + \infty} 2 \ln \left( \|A^n\|^{1/n} \right)= 2 \ln(\rho(A))=0,$$ where $\rho(A)=1$ is the spectral radius of $A$. So $h(T)=0$.