[proofplan] We first verify the logarithmic integrability hypotheses needed for Oseledets theorem for the invertible derivative cocycle. The substantive entropy input is the standard external Ergodic Margulis-Ruelle Inequality for $C^1$ compact Riemannian diffeomorphisms, stated explicitly below and applied only to ergodic invariant measures. For a general invariant measure, we use the [ergodic decomposition theorem](/theorems/3453) together with the affinity of metric entropy and the fiberwise Oseledets spectra to integrate the ergodic bound. Finally, in the ergodic case, invariance of the Oseledets data makes the spectrum constant almost everywhere. [/proofplan]

[step:Verify logarithmic integrability for the derivative cocycle]For each $n \in \mathbb{N}_0$, define the derivative cocycle map $A_n$ by \begin{align*} A_n: M \to \bigsqcup_{x \in M}\mathcal{L}(T_xM,T_{f^n(x)}M), \qquad x \mapsto df_x^n. \end{align*} The chain rule gives $A_{n+k}(x)=A_k(f^n(x))\circ A_n(x)$ for all $n,k \in \mathbb{N}_0$ and all $x \in M$. Since $f$ is $C^1$ and $M$ is compact, the function $x \mapsto \|df_x\|_{\mathrm{op}}$ is bounded. Since $f^{-1}$ is also $C^1$ and $M$ is compact, the function $x \mapsto \|d(f^{-1})_x\|_{\mathrm{op}}$ is bounded. Therefore \begin{align*} x \mapsto \log^+\|df_x\|_{\mathrm{op}} \end{align*} and \begin{align*} x \mapsto \log^+\|d(f^{-1})_x\|_{\mathrm{op}} \end{align*} belong to $L^1(M,\mathcal{B}(M),\mu)$. Hence the invertible derivative cocycle satisfies the logarithmic integrability assumptions of the [Oseledets Multiplicative Ergodic Theorem](/theorems/7764).[/step]

[guided]The theorem speaks about Lyapunov exponents of the derivative cocycle, so the first point is to check that the cocycle is in the range of Oseledets theorem. For each $n \in \mathbb{N}_0$, define \begin{align*} A_n: M \to \bigsqcup_{x \in M}\mathcal{L}(T_xM,T_{f^n(x)}M), \qquad x \mapsto df_x^n. \end{align*} This is the derivative cocycle because the chain rule gives \begin{align*} A_{n+k}(x)=A_k(f^n(x))\circ A_n(x) \end{align*} for all $n,k \in \mathbb{N}_0$ and all $x \in M$. The Oseledets theorem for an invertible linear cocycle requires integrability of the logarithmic size of both the forward cocycle generator and its inverse. The forward generator is $df_x:T_xM\to T_{f(x)}M$. The map $x\mapsto \|df_x\|_{\mathrm{op}}$ is continuous because $f$ is $C^1$ and the Riemannian metric varies smoothly. Since $M$ is compact, this [continuous function](/page/Continuous%20Function) is bounded. Thus \begin{align*} x \mapsto \log^+\|df_x\|_{\mathrm{op}} \end{align*} is bounded and therefore belongs to $L^1(M,\mathcal{B}(M),\mu)$ because $\mu$ is a probability measure. The inverse generator is controlled in the same way. Since $f$ is a $C^1$ diffeomorphism, $f^{-1}:M\to M$ is $C^1$. Hence $x\mapsto \|d(f^{-1})_x\|_{\mathrm{op}}$ is continuous, and compactness of $M$ makes it bounded. Therefore \begin{align*} x \mapsto \log^+\|d(f^{-1})_x\|_{\mathrm{op}} \end{align*} also belongs to $L^1(M,\mathcal{B}(M),\mu)$. These are exactly the logarithmic integrability hypotheses needed for the Oseledets Multiplicative Ergodic Theorem applied to the invertible derivative cocycle.[/guided]

[step:Record the Oseledets spectrum on a full-measure invariant set] By the Oseledets Multiplicative Ergodic Theorem applied to $(A_n)_{n\in\mathbb{N}_0}$, there is an $f$-invariant Borel set $M_0\subset M$ with $\mu(M_0)=1$ such that for every $x\in M_0$ there are an integer $r(x)\in\{1,\dots,d\}$, [real numbers](/page/Real%20Numbers) $\lambda_1(x)>\cdots>\lambda_{r(x)}(x)$, and nonzero subspaces $E_i(x)\subset T_xM$ satisfying \begin{align*} T_xM=E_1(x)\oplus\cdots\oplus E_{r(x)}(x). \end{align*} With $m_i(x):=\dim E_i(x)$, every $v\in E_i(x)\setminus\{0\}$ satisfies \begin{align*} \lim_{n\to\infty}\frac{1}{n}\log |df_x^n(v)|=\lambda_i(x). \end{align*} Changing $r$, $\lambda_i$, and $m_i$ on $M\setminus M_0$ does not change any integral with respect to $\mu$. [/step]

[step:Apply the ergodic Margulis-Ruelle inequality on ergodic components] We use the following standard external theorem, the Ergodic Margulis-Ruelle Inequality: if $\nu$ is an ergodic $f$-invariant Borel probability measure on a compact smooth Riemannian manifold and $f$ is a $C^1$ diffeomorphism, then the Oseledets exponents $\alpha_1>\cdots>\alpha_s$ and multiplicities $q_1,\dots,q_s$ of the derivative cocycle are $\nu$-almost everywhere constant and satisfy \begin{align*} h_\nu(f)\leq \sum_{1\leq j\leq s,\, \alpha_j>0} q_j\alpha_j. \end{align*} Let $\mathcal{P}_f(M)$ be the set of $f$-invariant Borel probability measures on $M$, and let $\mathcal{E}_f(M)\subset\mathcal{P}_f(M)$ be the subset of ergodic measures. By the ergodic decomposition theorem, there is a Borel probability measure $\tau$ on $\mathcal{E}_f(M)$ such that \begin{align*} \mu(B)=\int_{\mathcal{E}_f(M)}\nu(B)\,d\tau(\nu) \end{align*} for every Borel set $B\subset M$. The same theorem gives the entropy decomposition \begin{align*} h_\mu(f)=\int_{\mathcal{E}_f(M)} h_\nu(f)\,d\tau(\nu). \end{align*} For $\tau$-almost every ergodic component $\nu$, the hypotheses of the Ergodic Margulis-Ruelle Inequality are satisfied: $M$ is compact and Riemannian, $f$ is a $C^1$ diffeomorphism, and $\nu$ is an ergodic $f$-invariant Borel probability measure. Therefore \begin{align*} h_\nu(f)\leq \sum_{1\leq j\leq s(\nu),\, \alpha_j(\nu)>0} q_j(\nu)\alpha_j(\nu) \end{align*} for $\tau$-almost every $\nu\in\mathcal{E}_f(M)$. [/step]

[step:Integrate the ergodic bounds to obtain the non-ergodic inequality]For $x\in M_0$, let \begin{align*} \Phi(x):=\sum_{1\leq i\leq r(x),\, \lambda_i(x)>0}m_i(x)\lambda_i(x). \end{align*} Extend $\Phi$ arbitrarily to $M\setminus M_0$. On each ergodic component $\nu$, the Oseledets spectrum for $\nu$ agrees with the restriction of the same derivative cocycle to a $\nu$-full set, so the constant ergodic quantity satisfies \begin{align*} \Phi(x)=\sum_{1\leq j\leq s(\nu),\, \alpha_j(\nu)>0}q_j(\nu)\alpha_j(\nu) \end{align*} for $\nu$-almost every $x\in M$. Hence \begin{align*} \int_M \Phi(x)\,d\nu(x)=\sum_{1\leq j\leq s(\nu),\, \alpha_j(\nu)>0}q_j(\nu)\alpha_j(\nu) \end{align*} for $\tau$-almost every $\nu$. Combining this identity with the entropy decomposition and the ergodic Margulis-Ruelle inequality gives \begin{align*} h_\mu(f)\leq \int_{\mathcal{E}_f(M)}\int_M \Phi(x)\,d\nu(x)\,d\tau(\nu). \end{align*} By the defining property of the ergodic decomposition, applied to the nonnegative measurable function $\Phi$, the right-hand side equals \begin{align*} \int_M \Phi(x)\,d\mu(x). \end{align*} Therefore \begin{align*} h_\mu(f)\leq \int_M \sum_{1\leq i\leq r(x),\, \lambda_i(x)>0}m_i(x)\lambda_i(x)\,d\mu(x). \end{align*}[/step]

[guided]The non-ergodic case is not obtained by pretending that the exponents are constant. Instead, we decompose $\mu$ into ergodic invariant measures and apply the ergodic inequality on each component. Define the measurable nonnegative function $\Phi:M\to[0,\infty]$ on the Oseledets full-measure set $M_0$ by \begin{align*} \Phi(x):=\sum_{1\leq i\leq r(x),\, \lambda_i(x)>0}m_i(x)\lambda_i(x). \end{align*} Changing $\Phi$ on $M\setminus M_0$ does not affect integration with respect to $\mu$ or with respect to almost every ergodic component in the ergodic decomposition of $\mu$. Let $\tau$ be the ergodic decomposition measure on the set $\mathcal{E}_f(M)$ of ergodic $f$-invariant Borel probability measures. The ergodic decomposition theorem gives two facts. First, for every nonnegative Borel function $G:M\to[0,\infty]$, \begin{align*} \int_M G(x)\,d\mu(x)=\int_{\mathcal{E}_f(M)}\int_M G(x)\,d\nu(x)\,d\tau(\nu). \end{align*} Second, metric entropy decomposes affinely: \begin{align*} h_\mu(f)=\int_{\mathcal{E}_f(M)}h_\nu(f)\,d\tau(\nu). \end{align*} We apply these statements with $G=\Phi$. For $\tau$-almost every ergodic component $\nu$, the measure $\nu$ is an ergodic $f$-invariant Borel probability measure on the same compact Riemannian manifold, and $f$ is the same $C^1$ diffeomorphism. Thus the Ergodic Margulis-Ruelle Inequality applies. If $\alpha_1(\nu)>\cdots>\alpha_{s(\nu)}(\nu)$ are the $\nu$-almost everywhere constant Lyapunov exponents and $q_j(\nu)$ are their multiplicities, it gives \begin{align*} h_\nu(f)\leq \sum_{1\leq j\leq s(\nu),\, \alpha_j(\nu)>0}q_j(\nu)\alpha_j(\nu). \end{align*} Because the derivative cocycle is the same cocycle, the Oseledets spectrum recorded by $\Phi$ agrees with this constant spectrum on a $\nu$-full set. Therefore \begin{align*} \int_M \Phi(x)\,d\nu(x)=\sum_{1\leq j\leq s(\nu),\, \alpha_j(\nu)>0}q_j(\nu)\alpha_j(\nu). \end{align*} Integrating the previous [entropy inequality](/theorems/6729) with respect to $\tau$ gives \begin{align*} h_\mu(f)=\int_{\mathcal{E}_f(M)}h_\nu(f)\,d\tau(\nu)\leq \int_{\mathcal{E}_f(M)}\int_M \Phi(x)\,d\nu(x)\,d\tau(\nu). \end{align*} Finally, the defining integration formula for the ergodic decomposition, applied to the nonnegative function $\Phi$, gives \begin{align*} \int_{\mathcal{E}_f(M)}\int_M \Phi(x)\,d\nu(x)\,d\tau(\nu)=\int_M \Phi(x)\,d\mu(x). \end{align*} Substituting the definition of $\Phi$ proves \begin{align*} h_\mu(f)\leq \int_M \sum_{1\leq i\leq r(x),\, \lambda_i(x)>0}m_i(x)\lambda_i(x)\,d\mu(x). \end{align*}[/guided]

[step:Specialize the integral inequality when the measure is ergodic] Assume that $\mu$ is ergodic. The Oseledets splitting is equivariant under the derivative cocycle, so $r(x)$, the ordered Lyapunov exponents, and the multiplicities are $f$-invariant [measurable functions](/page/Measurable%20Functions) on a full-measure invariant set. Since every measurable $f$-invariant real-valued or integer-valued function is constant $\mu$-almost everywhere under ergodicity, there are constants $r\in\{1,\dots,d\}$, $\lambda_1>\cdots>\lambda_r$, and $m_1,\dots,m_r$ such that $r(x)=r$, $\lambda_i(x)=\lambda_i$, and $m_i(x)=m_i$ for $\mu$-almost every $x\in M$. Substituting these constants into the non-ergodic inequality gives \begin{align*} h_\mu(f)\leq \int_M \sum_{1\leq i\leq r,\, \lambda_i>0}m_i\lambda_i\,d\mu(x). \end{align*} Since $\mu(M)=1$, the integral of this constant function equals the constant itself. Thus \begin{align*} h_\mu(f)\leq \sum_{1\leq i\leq r,\, \lambda_i>0}m_i\lambda_i. \end{align*} This proves the ergodic specialization and completes the proof. [/step]