The categorical entropy–spectral radius conjecture

Let $Y$ be a smooth proper variety over $\mathbb{C}$. Write ${\mathcal D}^b(Y)$ for its bounded derived category of coherent sheaves, and let ${\rm End}^{FM}({\mathcal D}^b(Y))$ denote the Fourier–Mukai endofunctors of this category. For $F\in {\rm End}^{FM}({\mathcal D}^b(Y))$, let $h(F)$ be its categorical entropy at $t=0$, let $H\!H_\bullet(F)$ be the induced $\mathbb{C}$-linear endomorphism of the Hochschild homology $H\!H_\bullet(Y)$, and let $\rho$ denote spectral radius. Categorical entropy–spectral radius conjecture. If $H\!H_\bullet(F)$ is invertible, then

$$h(F)=\log\rho(H\!H_\bullet(F)).$$

Moreover, if $Y$ is projective and $F$ is an autoequivalence, then, writing $[F]$ for the induced action on the numerical Grothendieck group,

$$h(F)=\log\rho([F]).$$

The conjecture was proposed as a categorical analogue of the Gromov–Yomdin theorem. The lower bound $h(F)\geq\log\rho([F])$ is known, but the corresponding upper bound fails in general for perfect derived categories of smooth proper differential graded algebras because of phantom categories; consequently, the conjecture as stated is refuted in general. The failure does not affect the results of the paper.