Beta functions

Definitions and mathematical statement

Let $P$ be a finite primitive stochastic matrix. For real $t$, let $P^{[t]}$ have entry $p_{ij}^t$ wherever $p_{ij}>0$ and zero otherwise. Its Perron eigenvalue \[ \beta_P(t)=\rho(P^{[t]})>0 \] is a real-analytic function of $t$, with $\beta_P(1)=1$. This distinguished eigenvalue is the usual Perron-branch form of the beta invariant of the mixing Markov shift presented by $P$.

The source packages the invariant algebraically. Choose multiplicatively independent positive generators $g_1,\ldots,g_s$ for the group generated by the nonzero entries of $P$. Express each entry as a Laurent monomial in $u_j=g_j^t$ and factor $q(z)=\det(zI-P^{[t]})$ over the fraction field of the resulting Laurent polynomial ring. Let $b(z)$ be the monic irreducible factor whose root branch is $\beta_P(t)$. The source calls $1/b(z)$ the beta function; this specifies its reciprocal-polynomial convention.

Characterize the beta invariants arising from all finite primitive stochastic matrices. Equivalently, give intrinsic conditions on the distinguished Perron branch with its algebraic relation, together with a realization by such a matrix. Matrix size and allowed positive probabilities are unrestricted. The full characteristic polynomial is not part of this invariant: irrelevant factors are discarded. A description limited to entropy, to $\beta_P(0)$, or to matrices with rational entries would answer a more restricted question.

Short English statement

Characterize the beta invariant carried by the distinguished Perron eigenvalue of a powered stochastic matrix.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600011 (AMR-045-0011), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boyle - Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 10.2. Original source indexed by AMR-045-0011. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.