The error term of the prime orbit theorem for expanding semiflows

TL;DR

This paper investigates the distribution of prime periodic orbits in certain expanding semiflows, providing an asymptotic formula with a bounded error term related to entropy and Lyapunov exponents.

Contribution

It offers a new asymptotic estimate for the count of prime periodic orbits in suspension semiflows, including a bound on the error term under generic conditions.

Findings

Asymptotic formula for prime orbit count with explicit error bound

Error term bounded by an exponential involving entropy and Lyapunov exponent

Results applicable to angle multiplying maps on the circle

Abstract

We consider suspension semiflows of an angle multiplying map on the circle and study the distributions of periods of their periodic orbits. Under generic conditions on the roof function, we give an asymptotic formula on the number $\pi(T)$ of prime periodic orbits with period $\le T$. The error term is bounded, at least, by \[ \exp((1-\frac{1}{4\lceil \chi_{\max}/h_{\mathrm{top}}\rceil}+\varepsilon) h_{\top} T)\qquad {in the limit $T\to \infty$} \] for arbitrarily small $\varepsilon>0$, where $h_{\mathrm{top}}$ and $\chi_{\max}$ are respectively the topological entropy and the maximal Lyapunov exponent of the semiflow.