Spectral decomposition and \(\Omega\)-stability of flows with expanding measures (Q2189818)

For a flow \(\phi\) on a compact metric space \(X\) and a point \(x\in X\) one denotes by \(\phi_{\mathbb{R}}(x)\) the orbit of \(x\). For any \(x\in X\) and a constant \(\delta>0\), one defines \(\Gamma^\phi_\delta(x):=\{y\in X\mid d(\phi_t(x),\phi_{c(t)}(y))\leq\delta\ \forall t\in\mathbb{R}\}\), for some continuous function \(c:\mathbb{R}\to\mathbb{R}\), called the {dynamic \(\delta\)-ball of \(\phi\) centered at \(x\)}. The authors define a variant of measure-expanding flows as follows: Definition. A Borel measure \(\mu\) on \(X\) is said to be expanding for \(\phi\) if there is \(\delta>0\) (called an expanding constant of \(\mu\)) such that \(\mu(\Gamma^\phi_\delta(x)\backslash \phi_{\mathbb{R}}(x))=0\) for all \(x\in X\). We say that a flow \(\phi\) is measure-expanding (resp., invariantly measure-expanding) if every Borel measure (resp., invariant Borel measure) is expanding for \(\phi\). In Section 3, the authors prove the following: Theorem. A flow \(\phi\) on a compact metric space \(X\) is invariantly measure-expanding on \(X\) if and only if it is invariantly measure-expanding on its chain recurrent set \(\mathrm{CR}(\phi)\). The authors also construct (Example 3.2) a flow \(\phi\) which is invariantly measure-expanding on its nonwandering set \(\Omega(\phi)\) but not invariantly measure-expanding on \(X\) (note that \(\Omega(\phi)\subset\mathrm{CR}(\phi)\)). In Section 4, they present a measurable version of Smale's spectral decomposition theorem for flows. Theorem. If a flow \(\phi\) on a compact metric space \(X\) is invariantly measure-expanding on its chain recurrent set \(\mathrm{CR}(\phi)\) and has the invariantly measure-shadowing property on \(\mathrm{CR}(\phi)\), then \(\phi\) has the spectral decomposition, i.e., the nonwandering set \(\Omega(\phi)\) is decomposed as a disjoint union of finitely many invariant and closed subsets \[ \Omega(\phi)=B_1\cup\dots\cup B_l \] such that \(\phi\) is topologically transitive on each \(B_i\) for \(1\leq i\leq l\). The authors also give an example (see Example 4.5) to show that an invariantly measure-expanding flow with the shadowing property does not have a spectral decomposition. As a corollary, the authors derive following topological version of Smale's spectral decomposition which revises Theorem 5 of [\textit{M. Komuro}, Monatsh. Math. 98, 219--253 (1984; Zbl 0545.58037)]: Corollary. If a flow \(\phi\) on a compact metric space \(X\) is expansive on its chain recurrent set \(\mathrm{CR}(\phi)\) and has the shadowing property on \(\mathrm{CR}(\phi)\), then \(\phi\) has the spectral decomposition. In Section 5 the following terminology is introduced: \begin{itemize} \item[(i)] \(\operatorname{PO}(X_t)\) denotes the set of periodic points of the flow \(X_t\) of a \(C^1\) vector field \(X\) on a \(C^\infty\) manifold \(M\). \item[(ii)] A compact invariant subset \(\Lambda\) of \(M\) is said to be hyperbolic for \(X_t\) if it exhibits a splitting \(T_xM=E^s_x\oplus\langle X(x)\rangle\oplus E^u_x(x\in\Lambda)\) and there are constants \(C\geq 1\), \(0<\lambda<1\) such that the tangent flow \(DX_t:TM\to TM\) leaves the continuous invariant splitting and \[ \Vert DX_t|_{E^s_x}\Vert\leq Ce^{-\lambda t}\text{ and }\Vert DX_{-t}|_{E^u_x}\Vert\leq Ce^{-\lambda t} \] for \(x\in\Lambda\) and \(t\geq 0\). \item[(iii)] A \(C^1\) vector field \(X\in\mathcal{X}^1(M)\) is said to be Anosov if \(M\) is hyperbolic for \(X_t\). \item[(iv)] A \(C^1\) vector field \(X\in\mathcal{X}^1(M)\) is said to satisfies Axiom A if \(\Omega(X_t)\) is hyperbolic and \(\operatorname{PO}(X_t)\) is dense in \(\Omega(X_t)\smallsetminus\mathrm{Sing}(X)\), where Sing\((X)\) denotes the set of zeros of \(X\). \item[(v)] A vector field \(X\in\mathcal{X}^1(M)\) (or its integrated flow \(X_t\)) is said to be \(\Omega\)-stable if small perturbations of \(X\) preserve the topological structure of \(\Omega(X_t)\). \item[(vi)] The integrated flow \(X_t\) of \(X\in\mathcal{X}^1(M)\) is said to be \(C^1\) stably invariantly measure-expanding if there is a \(C^1\) neighborhood \(\mathcal{U}\) of \(X\) such that the integrated flow \(Y_t\) of \(Y\in\mathcal{U}\) is invariantly measure-expanding. \end{itemize} The authors characterize the measure-expanding flows on a compact \(C^\infty\) manifold by using the notion of \(\Omega\)-stability and prove the following theorem. Theorem. The integrated flow \(X_t\) of a nowhere vanishing \(C^1\) vector field \(X\in\mathcal{X}^1(M)\) is \(C^1\) stably invariantly measure-expanding if and only if \(X\) is \(\Omega\)-stable.