通过维数至少为3的闭流形的光滑微分同胚实现指定熵函数
Realizing prescribed entropy functions by smooth diffeomorphisms of closed manifolds of dimension at least three
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中文总结 AI 辅助
该研究针对维数≥3的闭光滑流形，构造满足指定熵函数的光滑微分同胚，得到Katok中间熵猜想的反例，证明中间熵性质在同痕于恒等映射的微分同胚中不是C^∞开的。
中文摘要 AI 辅助
设M是维数d≥3的闭光滑流形，给定一个紧可度量化的Choquet单形𝒮和𝒮上的有界非负仿射上半连续函数𝔢，我们构造M的一个C^∞微分同胚h，它同痕于id_M且支集在嵌入的d维实心环面D^{d-1}×S^1中，具有一个孤立的极小不变康托集K。h|_K的不变测度单形仿射同胚于带有熵函数𝔢的𝒮，而每个不支撑在K上的遍历h-不变测度都是固定点处的狄拉克测度。因此，遍历h-不变概率测度的测度论熵集合和h的拓扑熵为：ℋ_e(h)={0}∪𝔢(ex𝒮)，h_top(h)=max_{p∈𝒮}𝔢(p)。该映射h可被同痕于id_M的零熵微分同胚C^∞逼近。取𝒮为单点集时，可在每个此类M上得到Katok中间熵猜想的反例；我们还构造了此类反例h_j和数c_j>0，满足h_j在C^∞下趋于id_M，c_j趋于0，且ℋ_e(h_j)={0,c_j}，h_top(h_j)=c_j，由此说明中间熵性质在同痕于恒等映射的微分同胚中不是C^∞开的。
英文摘要
Let $M$ be a closed smooth manifold of dimension $d\geq3$. Given a compact metrizable Choquet simplex $\mathscr S$ and a bounded nonnegative affine upper semicontinuous function $\mathfrak e$ on $\mathscr S$, we construct a $C^\infty$ diffeomorphism $h$ of $M$, isotopic to $\operatorname{id}_M$ and supported in an embedded $d$-dimensional solid torus $D^{d-1}\times S^1$, with an isolated minimal invariant Cantor set $K$. The invariant-measure simplex of $h|_K$ is affinely homeomorphic to $\mathscr S$ with entropy function $\mathfrak e$, whereas every ergodic $h$-invariant measure not supported on $K$ is a Dirac measure at a fixed point. Consequently, the set of measure-theoretic entropies of ergodic $h$-invariant probability measures and the topological entropy of $h$ are \[ \mathscr H_{\mathrm e}(h)=\{0\}\cup\mathfrak e(\operatorname{ex}\mathscr S), \qquad h_{\mathrm{top}}(h)=\max_{p\in\mathscr S}\mathfrak e(p). \] The map $h$ is $C^\infty$-approximable by zero-entropy diffeomorphisms isotopic to $\operatorname{id}_M$. Taking $\mathscr S$ to be a singleton yields counterexamples to Katok's intermediate-entropy conjecture on every such $M$. We also construct such counterexamples $h_j$ and numbers $c_j>0$ with $h_j\to\operatorname{id}_M$ in $C^\infty$, $c_j\to0$, and \[ \mathscr H_{\mathrm e}(h_j)=\{0,c_j\}, \qquad h_{\mathrm{top}}(h_j)=c_j. \] Hence the intermediate-entropy property is not $C^\infty$ open among diffeomorphisms isotopic to the identity.
