Shub's entropy conjecture for C1 self-maps

Theorem 1.1: for some $q\ge1$ there is a $C^1$ self-map $f$ of $M=(\mathbb R/100\mathbb Z)\times(S^2)^{q+1}$ with $h_{\mathrm{top}}(f)=0$ while $f_*$ on $H_2(M;\mathbb R)$ has eigenvalue 2, so $h_{\mathrm{top}}(f)<\log\rho(f_*)$. The map is noninvertible, $\dim M=2q+3\ge5$, and the eigenvalue sits in intermediate degree, consistent with the known first-homology, top-degree and $C^\infty$ positive results. It does not address $C^1$ diffeomorphisms, $C^r$ maps for $r\ge2$, or the conjecture in low dimensions.

Theorem 1.1: for some $q\ge1$ there is a $C^1$ self-map $f$ of $M=(\mathbb R/100\mathbb Z)\times(S^2)^{q+1}$ with $h_{\mathrm{top}}(f)=0$ while $f_*$ on $H_2(M;\mathbb R)$ has eigenvalue 2, so $h_{\mathrm{top}}(f)<\log\rho(f_*)$. The map is noninvertible, $\dim M=2q+3\ge5$, and the eigenvalue sits in intermediate degree, consistent with the known first-homology, top-degree and $C^\infty$ positive results. It does not address $C^1$ diffeomorphisms, $C^r$ maps for $r\ge2$, or the conjecture in low dimensions.