Volume growth and topological entropy of partially hyperbolic systems (Q6165172)

Let \(f\) be a \(C^1\) diffeomorphism on a compact Riemannian manifold \(M\), and denote by \(E \oplus F\) the direct sum of \(Df\)-invariant bundles \(E\) and \(F\) on \(M\). We say that \(f\) admits a dominated splitting \(E \oplus F\), denoted by \(E \oplus_\prec F\), if there are constants \(C > 0\) and \(\lambda \in (0, 1)\) such that for any \(k \in \mathbb{N}\), any \(x \in M\) and any nonzero vectors \(v_E \in E(x)\), \(v_F \in F(x)\), we have \(\| D_xf^k (v_E) \| / \| v_E \| \leq C \lambda^k \cdot \| D_xf^k(v_F) \| / \| v_F \|\). We say that \(f\) admits a partially hyperbolic splitting \(TM = E^s \oplus E^1 \oplus E^2 \oplus \cdots \oplus E^l \oplus E^u\) if: (1) \(TM = (E^s \oplus E^1 \oplus \cdots \oplus E^i) \oplus_\prec (E^{i+1} \oplus \cdots \oplus E^l \oplus E^u)\) is a dominated splitting for \(0 \leq i \leq l\); and (2) \(E^s\) is uniformly contracting and \(E^u\) is uniformly expanding, that is, there are constants \(C > 0\) and \(\lambda \in (0, 1)\) such that for any \(k \in \mathbb{N}\), any \(x \in M\) and any non-zero vectors \(v^s \in E^s(x)\), \(v^u \in E^u(x)\), we have \(\| D_xf^k (v^s) \| / \| v^s \| \leq C \lambda^k\), \(\| D_xf^{-k} (v^u) \| / \| v^u \| \leq C \lambda^k\). The partially hyperbolic splitting is denoted by \(TM = E^s \oplus_\prec E^1 \oplus_\prec E^2 \oplus_\prec \cdots \oplus_\prec E^l \oplus_\prec E^u\) for the sake of simplicity. Let \((D_xf^n)^\wedge\) be the induced (by \(D_xf^n\)) map between exterior algebras of the tangent spaces \(T_xM\) and \(T_{f^n(x)}M\), where \(\| \cdot \|\) is the norm on operators induced from the Riemannian metric. The authors prove that if \(f\) admits a partially hyperbolic splitting \(TM = E^s \oplus_\prec E^1 \oplus_\prec E^2 \oplus_\prec \cdots \oplus_\prec E^l \oplus_\prec E^u\) such that \(\dim E^i = 1\) for \(1 \leq i \leq l\), then \(h_{\text{top}}(f) = \lim_{n \to +\infty} \frac{1}{n} \log \int \| (D_xf^n)^\wedge \| dx\). The equality is proved by showing that for sufficiently many points \(x \in M\) and a sufficiently small number \(\delta > 0\), \(\int_{B(x, n, \delta)} \| (D_zf^n)^\wedge \| dz \approx\) sub-exponentially small. As explained in the text, usually, the above estimate of the volume growth on dynamical balls is established only for some Lyapunov regular points. But the corresponding estimation on Lyapunov regular points is not sufficient (the Lyapunov regular points might have zero Lebesgue measure) to get an entropy formula (the equality). The authors first obtain a uniform bound on the volume growth on the stronger direction (with respect to a dominated splitting) on the dynamical balls at all points in \(M\), and then, verify that the maximal volume growth can be achieved by the stronger direction not only for Lyapunov regular points but also for all points in \(M\). This fact eventually leads us to the entropy formula.