# Katok & Hasselblatt — Candidate extraction (raw, pre-dedup)

## SCOUT: existing graph coverage (DS & Ergodic), node ids already present

### Ergodic theory & entropy
s_birkhoff_ergodic_theorem, s_von_neumann_mean_ergodic, s_wiener_ergodic_l1, s_kingman_subadditive_ergodic, s_oseledets_multiplicative_ergodic, s_oseledec_multiplicative_ergodic, s_pesin_entropy_formula, s_variational_principle_entropy, s_topological_entropy_h_top, s_measure_entropy_bounded_by_h_top, s_ergodic_decomposition_theorem, s_ks_entropy_invariance, s_manning_volume_entropy, s_volume_entropy_h_vol, s_topological_entropy_of_geodesic_flow

### Hyperbolic & smooth dynamics
s_anosov_closing_lemma, s_anosov_diffeomorphism, s_anosov_shadowing_lemma, s_smale_horseshoe, s_smale_birkhoff_homoclinic, s_smale_spectral_decomposition, s_axiom_A_diffeomorphism, s_srb_measure_existence, s_axiom_A_attractor, s_axiom_A_basic_set, s_mane_sad_sullivan_stability, s_mane_hyperbolicity_theorem, s_hyperbolic_fixed_point_of_diffeomorphism, s_smooth_diffeomorphism_with_nonzero_lyapunov_spectrum, s_pesin_set_theorem, s_pesin_stable_manifold_theorem

### Lyapunov exponents & stability
s_lyapunov_growth_function, s_lyapunov_spectrum_a_e, s_lyapunov_exponents_ae, s_furstenberg_kesten_theorem, s_poincare_linearization_theorem, s_lyapunov_stability_theorem, s_lyapunov_function_V, s_siegel_linearization

### Topological dynamics
s_poincare_recurrence, s_khinchin_recurrence, s_poincare_recurrence_theorem, s_furstenberg_multiple_recurrence, s_birkhoff_minimal_recurrence, s_minimal_subsystem, s_minimal_invariant_subset_M, s_subshift_of_finite_type, s_minimal_distal_topological_dynamical_system, s_furstenberg_distal_structure, s_equicontinuity_in_time_in_B, s_equicontinuity_implies_uniform_convergence

### Low-dimensional dynamics
s_denjoy_theorem, s_C2_circle_diffeomorphism_with_irrational_rotation_number, s_smooth_circle_diffeomorphism_with_diophantine_rotation_number, s_sharkovskii_theorem, s_feigenbaum_universality, s_kam_theorem, s_kam_theorem_2, s_kam_torus, s_rotation_number_of_immersion, s_sinai_billiard_ergodicity, s_bunimovich_stadium_ergodicity, s_peixoto_structural_stability

### Measure-theoretic foundations
s_ergodic_measure_preserving_system, s_measure_preserving_homeomorphism_of_compact_metric_space, s_ergodic_measure_preserving_transformation_with_finite_entropy, s_aperiodic_measure_preserving_transformation, s_weak_mixing, s_strong_mixing, s_mixing_transformation, s_two_bernoulli_shifts_of_equal_entropy, s_topological_conjugacy_to_bernoulli_shift, s_invariant_sigma_algebra

### Pressure & equilibrium states
s_bowen_specification_unique_equilibrium, s_topological_pressure_P_phi, s_bowen_partition_function_Z_n, s_pressure_function_of_s, s_convergence_to_equilibrium_state, s_gibbs_variational_principle, s_pressure_attains_supremum_at_gibbs_state, s_bowen_dimension_formula, s_mane_bowen_formula, s_bowen_periodic_equidistribution

### Chaos & bifurcations
s_diffeomorphism_with_transverse_homoclinic_point, s_transverse_homoclinic_orbit, s_homoclinic_orbit, s_heteroclinic_orbit, s_newhouse_phenomenon, s_C2_diffeomorphism_with_homoclinic_tangency, s_poincare_homoclinic_tangle, s_arnold_diffusion

### Techniques & auxiliary
t_ergodic_correspondence, t_transference_bridge, t_thermodynamic_formalism, t_birkhoff_normal_form, s_mather_connecting_orbits, s_brin_katok_local_entropy, s_hopf_geodesic_flow_ergodicity, s_pugh_shub_stable_ergodicity, s_pesin_sinai_srb_existence, s_ruelle_perron_frobenius_theorem

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## EXTRACTOR-A: Topological / combinatorial dynamics (52 items)
Foundational defs: dynamical system/cascade (axiom), flow (axiom), orbit (axiom), fixed/periodic point (axiom), topological conjugacy (axiom), topological equivalence of flows (axiom).
Examples: circle rotation R_alpha (state), expanding map of circle E_m (state), doubling map (state), toral translation (state), linear toral automorphism / cat map (state), full shift sigma on Sigma_N (state), subshift of finite type / topological Markov chain (state), Smale horseshoe (state), gradient flow (state), suspension/special flow (technique).
Recurrence/transitivity: topological transitivity (axiom), minimality (axiom), topological mixing (axiom), nonwandering set Omega(f) (state), chain recurrence / eps-chains (state), Poincare recurrence (theorem), Birkhoff recurrence theorem (theorem), existence of minimal sets (theorem).
Entropy/orbit growth: topological entropy via spanning/separated sets Bowen-Dinaburg (axiom), topological entropy via open covers Adler-Konheim-McAndrew (axiom), invariance of top entropy under conjugacy (theorem), entropy of expanding maps/shifts h=log spectral radius (theorem), growth rate of periodic orbits p_n(f) (state), Artin-Mazur zeta function (state), rationality of zeta for SFT zeta=1/det(I-zA) (theorem), dynamical zeta function (state).
Linear-algebraic/spectral: Perron-Frobenius theorem nonnegative matrices (theorem), spectral decomposition for topological Markov chains (theorem).
Equicontinuity/distality: equicontinuity/isometric systems (axiom), distal and proximal pairs (axiom), Furstenberg structure theorem for distal systems (theorem).
Variational principle (statement) (theorem).
One-dimensional: Sharkovskii ordering (state), Sharkovskii theorem (theorem), Li-Yorke theorem period three implies chaos (theorem), kneading theory/kneading invariant (technique), itinerary/coding map (technique), Schwarzian derivative (state), Singer's theorem (theorem).
Circle homeos: rotation number Poincare (state), rationality of rotation number iff periodic orbit (theorem), Poincare classification theorem (theorem), Denjoy theorem (theorem), Denjoy example/counterexample (state), Denjoy inequality (theorem).
Structural stability: Morse-Smale system (state), structural stability topological (axiom).
Additional: Markov partition (technique), pseudo-orbit eps-orbit (state), wandering/recurrent dichotomy (state), word/cylinder set symbolic (state), conjugacy of doubling map and one-sided 2-shift (theorem), local product structure / strips horseshoe (technique).

## EXTRACTOR-B: Hyperbolic dynamics (68 items)
1 hyperbolic linear map (state), 2 stable/unstable subspaces linear (state), 3 adapted/Lyapunov norm (technique), 4 hyperbolic fixed/periodic point (state), 5 hyperbolic set uniformly hyperbolic (state), 6 cone (cone field) criterion for hyperbolicity (technique), 7 local product structure (state), 8 Hadamard-Perron theorem / stable manifold theorem (theorem), 9 graph transform method (technique), 10 local stable/unstable manifolds (state), 11 global stable/unstable manifolds (state), 12 Grobman-Hartman theorem maps (theorem), 13 Hartman-Grobman for flows (theorem), 14 inclination lemma (lambda-lemma) (theorem), 15 Anosov diffeomorphism (state), 16 Anosov flow (state), 17 expanding map (state), 18 structural stability of expanding maps Shub (theorem), 19 linear hyperbolic automorphism of torus / Anosov example (state), 20 pseudo-orbit eps-orbit (state), 21 shadowing lemma (theorem), 22 Anosov closing lemma (theorem), 23 expansiveness (state), 24 structural stability (state), 25 Anosov structural stability theorem (theorem), 26 robustness/openness of hyperbolicity (theorem), 27 Holder continuity of hyperbolic splitting (theorem), 28 closing lemma (Pugh) (theorem), 29 structural stability theorem Robbin-Robinson-Mane (theorem), 30 nonwandering set (state), 31 Axiom A diffeomorphism (state), 32 spectral decomposition theorem Smale (theorem), 33 basic set (state), 34 locally maximal/isolated hyperbolic set (state), 35 spectral decomposition into mixing cyclic pieces (theorem), 36 filtration (technique), 37 no-cycle condition (axiom), 38 Omega-stability theorem (theorem), 39 strong transversality condition (axiom), 40 Markov partition for hyperbolic sets (technique), 41 existence of Markov partitions Sinai Bowen (theorem), 42 symbolic coding by SFT (theorem), 43 topological Markov chain SFT (state), 44 homoclinic point (state), 45 transverse homoclinic point (state), 46 Smale-Birkhoff homoclinic theorem (theorem), 47 Smale horseshoe (state), 48 horseshoes from homoclinic tangles (technique), 49 attractor topological (state), 50 Smale solenoid (state), 51 DA derived from Anosov attractor (state), 52 Plykin attractor (state), 53 absolute continuity of stable/unstable foliations (theorem), 54 Hopf argument (technique), 55 ergodicity of volume-preserving Anosov systems / geodesic flows (theorem), 56 geodesic flow on negatively curved manifold is Anosov (theorem), 57 stable/unstable horocycle (horosphere) foliations (state), 58 Lyapunov exponents (state), 59 nonuniform hyperbolicity via nonvanishing Lyapunov exponents (state), 60 partially hyperbolic system (state), 61 Livschitz (Livsic) theorem on cohomological equations (theorem), 62 cohomological equation / coboundary (state), 63 dynamical zeta function Artin-Mazur/Ruelle (state), 64 rationality/meromorphy of zeta for Axiom A (theorem), 65 periodic-orbit counting growth and topological entropy (theorem), 66 continuity/persistence of hyperbolic splitting (theorem), 67 topological transitivity/mixing of Anosov on whole manifold (theorem), 68 Smale's differentiable dynamics program synthesis (technique).

## EXTRACTOR-C: Ergodic theory / entropy / thermodynamic formalism (52 items)
1 measure-preserving transformation (state), 2 invariant measure (state), 3 Krylov-Bogolyubov theorem (theorem), 4 Poincare recurrence theorem measure-theoretic (theorem), 5 Rokhlin (Halmos-Rokhlin) lemma (theorem), 6 measurable partition / Rokhlin theory of partitions (technique), 7 conditional measures (disintegration) (technique), 8 ergodicity (state), 9 Birkhoff pointwise ergodic theorem (theorem), 10 maximal ergodic theorem/inequality (theorem), 11 von Neumann mean ergodic theorem (theorem), 12 ergodic decomposition (theorem), 13 Hopf argument (technique), 14 unique ergodicity (state), 15 unique ergodicity criterion (theorem), 16 Weyl equidistribution theorem (theorem), 17 Koopman operator (technique), 18 mixing strong mixing (state), 19 weak mixing (state), 20 spectral characterization of mixing/weak mixing (theorem), 21 discrete (pure point) spectrum (state), 22 Halmos-von Neumann theorem (theorem), 23 rigidity (state), 24 rank (rank-one) systems (state), 25 entropy of a partition (state), 26 conditional entropy of partitions (state), 27 metric (Kolmogorov-Sinai) entropy (state), 28 Kolmogorov-Sinai generator theorem / Sinai generator theorem (theorem), 29 Shannon-McMillan-Breiman theorem (theorem), 30 Abramov formula (theorem), 31 K-system (Kolmogorov automorphism) (state), 32 Bernoulli automorphism (state), 33 Ornstein isomorphism theorem (theorem), 34 topological entropy (state), 35 variational principle for entropy (theorem), 36 measure of maximal entropy (state), 37 topological pressure (state), 38 variational principle for pressure (theorem), 39 equilibrium state (state), 40 Gibbs measure (state), 41 Ruelle-Perron-Frobenius transfer operator (technique), 42 Ruelle-Perron-Frobenius theorem (theorem), 43 Lyapunov exponents (state), 44 Oseledets multiplicative ergodic theorem (theorem), 45 subadditive ergodic theorem Kingman (theorem), 46 nonuniform hyperbolicity Pesin set (state), 47 stable/unstable manifold theorem nonuniform/Pesin (theorem), 48 absolute continuity of invariant foliations (theorem), 49 Margulis-Ruelle inequality (theorem), 50 Pesin entropy formula (theorem), 51 SRB measure (state), 52 Furstenberg multiple recurrence theorem (theorem).
