Realizing prescribed entropy functions by smooth diffeomorphisms of closed manifolds of dimension at least three

Abstract

Let $M$ be a closed smooth manifold of dimension $d\ge 3$. Given a compact metrizable Choquet simplex $\mathcal{S}$ and a bounded nonnegative affine upper semicontinuous function $\mathfrak{e}$ on $\mathcal{S}$, we construct a $C^{\infty }$ diffeomorphism $h$ of $M$, isotopic to ${\mathrm{id}}_M$ 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 $\mathcal{S}$ 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 ${\mathrm{id}}_M$. Taking $\mathcal{S}$ 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 $h_j\to {\mathrm{id}}_M$ in $C^{\infty }$, $c_j\to 0$, 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.

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