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27 PRIMA 2013 Abstractsis well-known that for each rational map f on Ĉ withdeg(f) ≥ 2, the Julia set J(f) is a non-empty perfect compactsubset of Ĉ (thus J(f) contains uncountably manypoints), f : J(f) → J(f) is chaotic (at least in the senseof Devaney), anddim H ({x ∈ Ĉ | lim infn→∞ 1 n log ‖Df n (x)‖ > 0}) > 0,where dim H denotes the Hausdorff dimension with respectto the spherical distance on Ĉ and ‖ · ‖ denotesthe norm of the derivative with respect to the sphericalmetric on Ĉ. However, we show that for generic i.i.d.random dynamical systems of complex polynomials, allof the following (1) and (2) holds.(1) For all points x ∈ Ĉ, the orbit of the Dirac measureδ x at x under the dual of the transition operator ofthe system converges to a periodic cycle of probabilitymeasures on Ĉ.(2) For all but countably many points x ∈ Ĉ, for almostevery sequence γ = (γ 1 , γ 2 , γ 3 , · · · ) of polynomials,the Lyapunov exponent along γ starting with x isnegative. More precisely, there exists a constantc < 0, which depends only on the system, such thatfor all but countably many points x ∈ Ĉ, for almostevery sequence γ = (γ 1 , γ 2 , γ 3 , · · · ) of polynomials,1χ(γ, x) := lim n→∞ log ‖D(γn ◦ · · · ◦nγ 1 )(x)‖ exists and χ(γ, x) ≤ c.Note that each of (1) and (2) cannot hold in the usualiteration dynamics of a single rational map f withdeg(f) ≥ 2. Therefore the picture of the randomcomplex dynamics is completely different fromthat of the usual complex dynamics. We remarkthat even the chaos of the random system disappears inC 0 sense, the chaos of the system may remain in C 1 sense,and we have to consider the “gradation between the nonchaoticityand the chaoticity”.References[1] H. Sumi, Random complex dynamics and semigroupsof holomorphic maps, Proc. London. Math. Soc. (2011),102 (1), 50–112.[2] H. Sumi, Cooperation principle, stability andbifurcation in random complex dynamics, preprint,http://arxiv.org/abs/1008.3995.The semi-classical zeta function for contactAnosov flowMasato TsujiiKyushu University, Japantsujii@math.kyushu-u.ac.jpWe consider the semi-classical (or Gutzwiller-Voros) zetafunction for a C ∞ contact Anosov flow. This is definedby the formula⎛⎞Z sc(s) = exp ⎝− ∑ ∞∑ e −s·m·|γ|⎠m · | det(Id − Dγ∈Γ m=0γ m)|1/2 where Γ is the set of prime periodic orbits. (|γ| and D γdenote the prime period and the transversal Jacobian ofγ ∈ Γ respectively. )Investigating the spectrum of transfer operators associatedto the flow, we prove that the zeros of Z sc(s) arecontained in the region{s ∈ C | |R(s)| < τ or R(s) < −χ 0 + τ}for arbitrarily small τ > 0 up to finitely many exceptions,where χ 0 is the hyperbolicity exponent of the flow. Furtherwe show that the zeros in the strip {s ∈ C | |R(s)|