Intro; Preface; Contents; Index of notations; 1 Introduction; 1.1 Statistical properties of chaotic differentiable dynamical systems; 1.2 Transfer operators. Dynamical determinants. Resonances; 1.3 Main results. Examples; 1.4 Main techniques; Comments; Part I Smooth expanding maps; 2 Smooth expanding maps: The spectrum of the transfer operator; 2.1 Transfer operators for smooth expanding maps on Hölder functions; 2.2 Transfer operators for smooth expanding maps on Sobolev spaces; 2.2.1 Isotropic Sobolev spaces Htp and good systems of charts

2.2.2 Bounding the essential spectral radius (Theorem 2.15)2.2.3 The key local Lasota-Yorke bound (Lemma 2.21); 2.2.4 Fragmentation and reconstitution: Technical lemmas; 2.3 The essential spectral radius on Sobolev spaces: Interpolation; 2.3.1 Complex interpolation; 2.3.2 Proof of Theorem 2.15 on Htp for integer differentiability; 2.4 The essential spectral radius: Dyadic decomposition; 2.4.1 A Paley-Littlewood description of Htp and Ct*; 2.4.2 Proof of Lemma 2.21 and Theorem 2.15: The general case; 2.5 Spectral stability and linear response à la Gouëzel-Keller-Liverani; Problems; Comments

3 Smooth expanding maps: Dynamical determinants3.1 Ruelle's theorem on the dynamical determinant; 3.1.1 Dynamical zeta functions; 3.2 Ruelle's theorem via kneading determinants; 3.2.1 Outline; 3.2.2 Flat traces; 3.3 Dynamical determinants: Completing the proof of Theorem 3.5; 3.3.1 Proof of Theorem 3.5 if >d+t; 3.3.2 Nuclear power decomposition via approximation numbers; 3.3.3 Asymptotic vanishing of flat traces of the non-compact term; 3.3.4 The case d+t of low differentiability; Problems; Comments; Part II Smooth hyperbolic maps; 4 Anisotropic Banach spaces defined via cones

4.1 Transfer operators for hyperbolic dynamics4.1.1 Hyperbolic dynamics and anisotropic spaces; 4.1.2 Bounding the essential spectral radius (Theorem 4.6); 4.1.3 Reducing to the transitive case; 4.2 The spaces Wp,*t,s and Wp,**t,s; 4.2.1 Charts and cone systems adapted to (T,V); 4.2.2 Formal definition of the spaces Wp,*t,s and Wp,**t,s; 4.3 The local Lasota-Yorke lemma and the proof of Theorem 4.6; 4.3.1 The Paley-Littlewood description of the spaces and the local Lasota-Yorke lemma; 4.3.2 Fragmentation, reconstitution, and the proof of Theorem 4.6; Problems; Comments

5 A variational formula for the essential spectral radius5.1 Yet another anisotropic Banach space: Bt,s; 5.1.1 Defining Bt,s; 5.2 Bounding the essential spectral radius on Bt,s (Theorem 5.1); 5.3 Spectral stability and linear response; Problems; Comments; 6 Dynamical determinants for smooth hyperbolic dynamics; 6.1 Dynamical determinants via regularised determinants and flat traces; 6.2 Proof of Theorem 6.2 on dT,g(z) if r-1> d+ t-s; 6.3 Theorem 6.2 in low differentiability r-1d+t-s; 6.4 Operators on vector bundles and dynamical zeta functions; Problems; Comments