Prime orbit theorems for expanding Thurston maps: Lattès maps and split Ruelle operators.(English)Zbl 1547.37035
Summary: We obtain an analog of the prime number theorem for a class of branched covering maps on the 2-sphere \(S^2\) called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumption. More precisely, we show that the number of primitive periodic orbits, ordered by a weight on each point induced by a non-constant (eventually) positive real-valued Hölder continuous function on \(S^2\) satisfying the \(\alpha\)-strong non-integrability condition, is asymptotically the same as the well-known logarithmic integral, with an exponential error bound. In particular, our results apply to postcritically-finite rational maps for which the Julia set is the whole Riemann sphere. Moreover, a stronger result is obtained for Lattès maps.
MSC:
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 37F15 | Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems |
| 37B10 | Symbolic dynamics |
| 11A41 | Primes |
| 11N05 | Distribution of primes |
| 11N45 | Asymptotic results on counting functions for algebraic and topological structures |
Keywords:
expanding Thurston map;postcritically-finite map;Lattès map;prime orbit theorem;Ruelle zeta function;split Ruelle operatorCite
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