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The error term in the prime orbit theorem for expanding semiflows

Published online by Cambridge University Press:  24 January 2017

MASATO TSUJII
MASATO TSUJII*
Affiliation:
Department of Mathematics, Kyushu University, Motooka 744, Nishi-ku, Fukuoka, 819-0395, Japan email tsujii@math.kyushu-u.ac.jp
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Abstract

We consider suspension semiflows of angle-multiplying maps 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 $\unicode[STIX]{x1D70B}(T)$ of prime periodic orbits with period $\leq T$. The error term is bounded, at least, by

$$\begin{eqnarray}\exp \biggl(\biggl(1-\frac{1}{4\lceil \unicode[STIX]{x1D712}_{\text{max}}/h_{\text{top}}\rceil }+\unicode[STIX]{x1D700}\biggr)h_{\text{top}}\cdot T\biggr)\quad \text{in the limit }T\rightarrow \infty\end{eqnarray}$$
for arbitrarily small $\unicode[STIX]{x1D700}>0$, where $h_{\text{top}}$ and $\unicode[STIX]{x1D712}_{\text{max}}$ are, respectively, the topological entropy and the maximal Lyapunov exponent of the semiflow.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 5 , August 2018 , pp. 1954 - 2000
Copyright
© Cambridge University Press, 2017 

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