Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps

TY - JOUR

T1 - Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps

AU - Baladi, Viviane

AU - Marmi, S.

AU - Sauzin, D.

PY - 2014/6

Y1 - 2014/6

N2 - For a piecewise expanding unimodal interval map f with unique absolutely continuous invariant probability measure μ, a perturbation X, and an observable φ, the susceptibility function is Ψphi;(z)= ∑ k=0 ∞ zk∫ X(x)φ'(f k)(x) (fk)'(x) dμ .Combining previous results [V. Baladi, On the susceptibility function of piecewise expanding interval maps. Comm. Math. Phys. 275 (2007), 839-859; V. Baladi and D. Smania, Linear response for piecewise expanding unimodal maps. Nonlinearity 21 (2008), 677-711] (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon [Natural boundaries and spectral theory. Adv. Math. 226 (2011), 4902-4920] (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon [Sur les séries de Dirichlet. Ann. Sci. Éc. Norm. Supér. (3) 66 (1949), 263-310]), we show that density of the postcritical orbit (a generic condition) implies that Ψ phi;(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functions of Ψphi;(z), associated with precritical orbits. If the perturbation X is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the non-tangential limit of Ψphi;(z) as z → 1 exists and coincides with the derivative of the absolutely continuous invariant probability measure with respect to the map ('linear response formula'). Applying the Wiener-Wintner theorem, we study the singularity type of non-tangential limits of Ψphi;(z) as z → eω for real ω. An additional 'law of the iterated logarithm' typicality assumption on the postcritical orbit gives stronger results.

AB - For a piecewise expanding unimodal interval map f with unique absolutely continuous invariant probability measure μ, a perturbation X, and an observable φ, the susceptibility function is Ψphi;(z)= ∑ k=0 ∞ zk∫ X(x)φ'(f k)(x) (fk)'(x) dμ .Combining previous results [V. Baladi, On the susceptibility function of piecewise expanding interval maps. Comm. Math. Phys. 275 (2007), 839-859; V. Baladi and D. Smania, Linear response for piecewise expanding unimodal maps. Nonlinearity 21 (2008), 677-711] (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon [Natural boundaries and spectral theory. Adv. Math. 226 (2011), 4902-4920] (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon [Sur les séries de Dirichlet. Ann. Sci. Éc. Norm. Supér. (3) 66 (1949), 263-310]), we show that density of the postcritical orbit (a generic condition) implies that Ψ phi;(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functions of Ψphi;(z), associated with precritical orbits. If the perturbation X is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the non-tangential limit of Ψphi;(z) as z → 1 exists and coincides with the derivative of the absolutely continuous invariant probability measure with respect to the map ('linear response formula'). Applying the Wiener-Wintner theorem, we study the singularity type of non-tangential limits of Ψphi;(z) as z → eω for real ω. An additional 'law of the iterated logarithm' typicality assumption on the postcritical orbit gives stronger results.

U2 - 10.1017/etds.2012.161

DO - 10.1017/etds.2012.161

M3 - Journal article

SN - 0143-3857

VL - 34

SP - 777

EP - 800

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 3

ER -