Entropy and Deligne periods

Definitions and mathematical statement

Fix a prime $p$. Let $\mathbb Q_p$ be the $p$-adic field and $\mathbb C_p$ the completion of its algebraic closure. Use the branch of the $p$-adic logarithm $\log_p:\mathbb C_p^*\to\mathbb C_p$ with $\log_p(p)=0$, agreeing near $1$ with \[ \log_p(1+u)=\sum_{n\geq1}\frac{(-1)^{n+1}u^n}{n}\qquad(|u|_p<1). \] For $\lambda\in\mathbb Q_p^*$ let $T_\lambda:\mathbb Q_p\to\mathbb Q_p$ be $T_\lambda(x)=\lambda x$. This restriction on $\lambda$ is necessary for the displayed map to be a nonzero multiplication self-map of $\mathbb Q_p$.

Can one define a meaningful $p$-adic entropy-like invariant $h_p$, on a suitable class of dynamical systems containing these maps, such that \[ h_p(T_\lambda)=\log_p\lambda\qquad(\lambda\in\mathbb Q_p^*)? \] The source suggests invariance under topological conjugacy as one possible desired property: if a homeomorphism $\psi$ conjugates $S$ to $T$, so $\psi\circ S=T\circ\psi$, then $h_p(S)=h_p(T)$. This is proposed as an example of an invariance requirement, not a complete supplied list of axioms. The target values are $p$-adic, rather than the nonnegative real values of ordinary entropy.

Specify the admissible system class and axioms that make such an invariant meaningful, and determine whether the normalization and desired invariances can coexist. The supplied summary associates this project with Deligne and $p$-adic periods, but gives no precise period identity or additional condition. Thus it is an invariant-construction problem whose full axioms are still to be formulated.

Short English statement

Seek a $p$-adic analogue of entropy whose value on multiplication by $\lambda$ on $\mathbb Q_p$ is $\log_p\lambda$. Identify a meaningful system class and invariance properties, such as an appropriate form of conjugacy invariance.

Sources

• Thomas Ward, Six problems in Algebraic Dynamics (2006), Problem F, https://www.imath.kiev.ua/~skolyada/kevin.pdf; accessed 2026-07-29, with PDF-text extraction recorded in the supplied provenance.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $4300006$, source code catalogue entry, Entropy and Deligne periods. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.
• The supplied summary mentions Deligne/$p$-adic periods and a Langlands/period interpretation but supplies no additional publication or precise mathematical relation.