[proofplan] We compute topological entropy from Bowen balls for the quotient metric on $\mathbb{T}^d$. Hyperbolicity gives a real stable-unstable splitting of $\mathbb{R}^d$, and the Bowen ball condition says that the initial unstable displacement must be exponentially small while the stable displacement remains uniformly bounded. A standard Bowen-ball volume comparison then shows that the minimal number of orbit balls of length $n$ needed to cover the torus grows like $|\det(A|_{E^u})|^n$. Finally, linear algebra identifies $|\det(A|_{E^u})|$ with the product of the moduli of the eigenvalues outside the unit circle. [/proofplan]

[step:Define Bowen balls for a quotient metric on the torus] Choose any Euclidean norm $|\cdot|$ on $\mathbb{R}^d$. Define the quotient metric $\rho: \mathbb{T}^d \times \mathbb{T}^d \to [0,\infty)$ by \begin{align*} \rho(x+\mathbb{Z}^d,y+\mathbb{Z}^d) := \min_{m \in \mathbb{Z}^d} |x-y-m|. \end{align*} Since $\mathbb{T}^d$ is compact, this metric induces its standard [quotient topology](/page/Quotient%20Topology). For $n \in \mathbb{N}$ and $\varepsilon>0$, define the Bowen metric $\rho_n: \mathbb{T}^d \times \mathbb{T}^d \to [0,\infty)$ by \begin{align*} \rho_n(p,q) := \max_{\{k \in \{0,\dots,n-1\}\}} \rho(f_A^k(p),f_A^k(q)). \end{align*} For $p \in \mathbb{T}^d$, define the length-$n$ Bowen ball of radius $\varepsilon$ centered at $p$ by \begin{align*} B_n(p,\varepsilon) := \{q \in \mathbb{T}^d : \rho_n(p,q)<\varepsilon\}. \end{align*} Let $r_n(\varepsilon)$ denote the smallest cardinality of a subset $F \subset \mathbb{T}^d$ such that \begin{align*} \mathbb{T}^d = \bigcup_{p \in F} B_n(p,\varepsilon). \end{align*} By the spanning-set definition of topological entropy, \begin{align*} h_{\mathrm{top}}(f_A)=\lim_{\varepsilon \downarrow 0}\limsup_{n\to\infty}\frac{1}{n}\log r_n(\varepsilon). \end{align*} This is the definition of topological entropy used below. [/step]

[step:Split Euclidean space into stable and unstable generalized eigenspaces] Let $A_{\mathbb{C}}: \mathbb{C}^d \to \mathbb{C}^d$ denote the complex-linear extension of $A$. For each eigenvalue $\lambda \in \mathbb{C}$, let $G_\lambda \subset \mathbb{C}^d$ denote the generalized eigenspace \begin{align*} G_\lambda := \ker((A_{\mathbb{C}}-\lambda I)^d). \end{align*} Define the complex unstable and stable subspaces by \begin{align*} E_{\mathbb{C}}^u := \bigoplus_{\{ \lambda : |\lambda|>1\}} G_\lambda, \quad E_{\mathbb{C}}^s := \bigoplus_{\{ \lambda : |\lambda|<1\}} G_\lambda. \end{align*} Because no eigenvalue has modulus $1$, the generalized eigenspace decomposition gives \begin{align*} \mathbb{C}^d = E_{\mathbb{C}}^s \oplus E_{\mathbb{C}}^u. \end{align*} Both subspaces are invariant under complex conjugation, since $A$ has real entries. Therefore they are complexifications of real $A$-invariant subspaces $E^s,E^u \subset \mathbb{R}^d$, and \begin{align*} \mathbb{R}^d = E^s \oplus E^u. \end{align*} We use the standard stable-unstable splitting theorem for hyperbolic linear maps: there are constants $C_s,C_u>0$ and numbers $\alpha_s,\alpha_u \in (0,1)$ such that, for every $n \in \mathbb{N}$, \begin{align*} |A^n v_s| \leq C_s \alpha_s^n |v_s| \quad \text{for every } v_s \in E^s \end{align*} and \begin{align*} |A^{-n} v_u| \leq C_u \alpha_u^n |v_u| \quad \text{for every } v_u \in E^u. \end{align*} (citing a result not yet in the wiki: Stable-unstable splitting for hyperbolic linear maps.) [/step]

[step:Compare Bowen-ball volume with unstable Jacobian growth] Let $\mu$ denote the normalized Haar probability measure on $\mathbb{T}^d$. Define the unstable Jacobian \begin{align*} J_u := |\det(A|_{E^u}:E^u \to E^u)|. \end{align*} Since $A(E^u)=E^u$ and all eigenvalues of $A|_{E^u}$ have modulus greater than $1$, we have $J_u>1$. We use the standard Bowen-ball volume comparison for linear toral automorphisms: there exist constants $\varepsilon_0>0$ and $c_1,c_2>0$, depending only on $A$ and the chosen metric $\rho$, such that for every $\varepsilon \in (0,\varepsilon_0]$, every $p \in \mathbb{T}^d$, and every $n \in \mathbb{N}$, \begin{align*} c_1(\varepsilon) J_u^{-n} \leq \mu(B_n(p,\varepsilon)) \leq c_2(\varepsilon) J_u^{-n}. \end{align*} Here $c_1(\varepsilon)$ and $c_2(\varepsilon)$ are positive constants independent of $p$ and $n$. The reason is that, after lifting to $\mathbb{R}^d$ below the injectivity radius of the quotient map, the Bowen condition requires the unstable component of the initial displacement to lie in a set whose $E^u$-volume is comparable to $J_u^{-n}$, while the stable component remains confined to a fixed bounded set. This is precisely the Bowen-ball volume comparison for linear toral automorphisms. (citing a result not yet in the wiki: Bowen-ball volume comparison for linear toral automorphisms.) [/step]

[step:Convert Bowen-ball volume estimates into entropy bounds]Fix $\varepsilon \in (0,\varepsilon_0]$. If $F \subset \mathbb{T}^d$ is an $(n,\varepsilon)$-spanning set, then the Bowen balls $\{B_n(p,\varepsilon):p \in F\}$ cover $\mathbb{T}^d$. Using subadditivity of the measure $\mu$ and the upper Bowen-ball volume estimate, \begin{align*} 1=\mu(\mathbb{T}^d) \leq \sum_{p \in F}\mu(B_n(p,\varepsilon)) \leq |F| c_2(\varepsilon)J_u^{-n}. \end{align*} Taking the infimum over all such spanning sets $F$ gives \begin{align*} r_n(\varepsilon) \geq c_2(\varepsilon)^{-1}J_u^n. \end{align*} Therefore \begin{align*} \liminf_{n\to\infty}\frac{1}{n}\log r_n(\varepsilon) \geq \log J_u. \end{align*} For the reverse inequality, let $E_n \subset \mathbb{T}^d$ be a maximal subset with the property that $\rho_n(p,q)\geq \varepsilon/2$ whenever $p,q \in E_n$ and $p \neq q$. Maximality implies that $E_n$ is an $(n,\varepsilon/2)$-spanning set, hence also an $(n,\varepsilon)$-spanning set. The Bowen balls $\{B_n(p,\varepsilon/4):p \in E_n\}$ are pairwise disjoint. Using additivity of $\mu$ on disjoint measurable sets and the lower Bowen-ball volume estimate, \begin{align*} 1=\mu(\mathbb{T}^d) \geq \sum_{p \in E_n}\mu(B_n(p,\varepsilon/4)) \geq |E_n| c_1(\varepsilon/4)J_u^{-n}. \end{align*} Thus \begin{align*} r_n(\varepsilon) \leq |E_n| \leq c_1(\varepsilon/4)^{-1}J_u^n. \end{align*} Consequently \begin{align*} \limsup_{n\to\infty}\frac{1}{n}\log r_n(\varepsilon) \leq \log J_u. \end{align*} Combining the lower and upper bounds yields \begin{align*} \lim_{n\to\infty}\frac{1}{n}\log r_n(\varepsilon)=\log J_u \end{align*} for every sufficiently small $\varepsilon>0$. Taking $\varepsilon \downarrow 0$ in the spanning-set formula for entropy gives \begin{align*} h_{\mathrm{top}}(f_A)=\log J_u. \end{align*}[/step]

[guided]The volume estimate says that every length-$n$ Bowen ball has measure comparable to $J_u^{-n}$. We now turn that geometric fact into an entropy computation. The bridge is simple: if sets of size about $J_u^{-n}$ cover a probability space of total measure $1$, then at least about $J_u^n$ of them are needed; conversely, a maximal separated family produces a cover, and disjoint smaller Bowen balls bound its size from above. Fix $\varepsilon \in (0,\varepsilon_0]$. Let $F \subset \mathbb{T}^d$ be an $(n,\varepsilon)$-spanning set, meaning that \begin{align*} \mathbb{T}^d = \bigcup_{p \in F} B_n(p,\varepsilon). \end{align*} Since $\mu$ is a probability measure, $\mu(\mathbb{T}^d)=1$. Since measure is subadditive over countable unions and $F$ is finite for a minimal spanning set on the compact [metric space](/page/Metric%20Space) $\mathbb{T}^d$, we get \begin{align*} 1=\mu(\mathbb{T}^d) \leq \sum_{p \in F}\mu(B_n(p,\varepsilon)). \end{align*} The upper Bowen-ball volume estimate gives $\mu(B_n(p,\varepsilon)) \leq c_2(\varepsilon)J_u^{-n}$ for every $p \in F$, with a constant independent of $p$ and $n$. Therefore \begin{align*} 1 \leq |F| c_2(\varepsilon)J_u^{-n}. \end{align*} Solving this inequality for $|F|$ gives \begin{align*} |F| \geq c_2(\varepsilon)^{-1}J_u^n. \end{align*} Since $r_n(\varepsilon)$ is the smallest possible cardinality of such a spanning set, the same lower bound holds for $r_n(\varepsilon)$: \begin{align*} r_n(\varepsilon) \geq c_2(\varepsilon)^{-1}J_u^n. \end{align*} Taking logarithms, dividing by $n$, and letting $n \to \infty$, the fixed multiplicative constant disappears: \begin{align*} \liminf_{n\to\infty}\frac{1}{n}\log r_n(\varepsilon) \geq \log J_u. \end{align*} For the upper bound, choose $E_n \subset \mathbb{T}^d$ maximal with respect to the property that distinct points are separated in the Bowen metric at scale $\varepsilon/2$: \begin{align*} \rho_n(p,q)\geq \varepsilon/2 \quad \text{whenever } p,q \in E_n \text{ and } p \neq q. \end{align*} Maximality means that no point of $\mathbb{T}^d$ can be added while preserving this separation. Hence every point of $\mathbb{T}^d$ lies within $\rho_n$-distance less than $\varepsilon/2$ of some point of $E_n$, so $E_n$ is an $(n,\varepsilon/2)$-spanning set. In particular it is also an $(n,\varepsilon)$-spanning set, and therefore \begin{align*} r_n(\varepsilon) \leq |E_n|. \end{align*} Now distinct points of $E_n$ have $\rho_n$-distance at least $\varepsilon/2$. Therefore the smaller Bowen balls $B_n(p,\varepsilon/4)$ centered at points $p \in E_n$ are pairwise disjoint: if two such balls met, the triangle inequality for $\rho_n$ would put their centers within distance less than $\varepsilon/2$. Additivity of $\mu$ over pairwise disjoint measurable sets gives \begin{align*} 1=\mu(\mathbb{T}^d) \geq \sum_{p \in E_n}\mu(B_n(p,\varepsilon/4)). \end{align*} The lower Bowen-ball volume estimate gives $\mu(B_n(p,\varepsilon/4)) \geq c_1(\varepsilon/4)J_u^{-n}$ for every $p \in E_n$. Hence \begin{align*} 1 \geq |E_n| c_1(\varepsilon/4)J_u^{-n}. \end{align*} Solving for $|E_n|$ yields \begin{align*} |E_n| \leq c_1(\varepsilon/4)^{-1}J_u^n. \end{align*} Since $r_n(\varepsilon) \leq |E_n|$, we obtain \begin{align*} r_n(\varepsilon) \leq c_1(\varepsilon/4)^{-1}J_u^n. \end{align*} Taking logarithms, dividing by $n$, and letting $n \to \infty$ gives \begin{align*} \limsup_{n\to\infty}\frac{1}{n}\log r_n(\varepsilon) \leq \log J_u. \end{align*} The lower and upper estimates match, so \begin{align*} \lim_{n\to\infty}\frac{1}{n}\log r_n(\varepsilon)=\log J_u. \end{align*} Finally, the definition of topological entropy is obtained by taking the limit as the radius tends to $0$: \begin{align*} h_{\mathrm{top}}(f_A)=\lim_{\varepsilon \downarrow 0}\limsup_{n\to\infty}\frac{1}{n}\log r_n(\varepsilon)=\log J_u. \end{align*}[/guided]

[step:Identify the unstable determinant with the product of expanding eigenvalue moduli] It remains to compute $J_u$. The complexification of $A|_{E^u}:E^u \to E^u$ is precisely the restriction of $A_{\mathbb{C}}$ to \begin{align*} E_{\mathbb{C}}^u=\bigoplus_{\{ \lambda : |\lambda|>1\}}G_\lambda. \end{align*} On each generalized eigenspace $G_\lambda$, the matrix of $A_{\mathbb{C}}$ has the single eigenvalue $\lambda$ repeated $\dim_{\mathbb{C}}G_\lambda$ times, counted with algebraic multiplicity. Therefore the determinant of $A_{\mathbb{C}}|_{E_{\mathbb{C}}^u}$ is \begin{align*} \det(A_{\mathbb{C}}|_{E_{\mathbb{C}}^u})=\prod_{\{i \in \{1,\dots,d\}:|\lambda_i|>1\}}\lambda_i. \end{align*} Because $A|_{E^u}$ is a real [linear map](/page/Linear%20Map) and complexification preserves determinant, \begin{align*} \det(A|_{E^u})=\det(A_{\mathbb{C}}|_{E_{\mathbb{C}}^u}). \end{align*} Taking absolute values gives \begin{align*} J_u=|\det(A|_{E^u})|=\prod_{\{i \in \{1,\dots,d\}:|\lambda_i|>1\}}|\lambda_i|. \end{align*} Thus \begin{align*} \log J_u=\sum_{\{i \in \{1,\dots,d\}:|\lambda_i|>1\}}\log |\lambda_i|. \end{align*} Combining this identity with $h_{\mathrm{top}}(f_A)=\log J_u$ proves the stated entropy formula. [/step]