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We prove the topological (or combinatorial) rigidity property for real polynomials with all critical points real and nondegenerate, which completes the last step in solving the density of Axiom A conjecture in real one-dimensional dynamics. Contents 1. Introduction 1.1. Statement of results 1.2. Organization of this work 1.3. General terminologies and notation 2. Density of Axiom A follows from the Rigidity Theorem 3. Derivation of the Rigidity Theorem from the Reduced Rigidity Theorem | Annals of Mathematics Rigidity for real polynomials By O. Kozlovski W. Shen andS. van Strien Annals of Mathematics 165 2007 749 841 Rigidity for real polynomials By O. Kozlovski W. Shen and S. VAN Strien Abstract We prove the topological or combinatorial rigidity property for real polynomials with all critical points real and nondegenerate which completes the last step in solving the density of Axiom A conjecture in real one-dimensional dynamics. Contents 1. Introduction 1.1. Statement of results 1.2. Organization of this work 1.3. General terminologies and notation 2. Density of Axiom A follows from the Rigidity Theorem 3. Derivation of the Rigidity Theorem from the Reduced Rigidity Theorem 4. Statement of the Key Lemma 5. Yoccoz puzzle and the spreading principle 5.1. External angles 5.2. Yoccoz puzzle partition 5.3. Spreading principle 6. Reduction to the infinitely renormalizable case 6.1. A real partition 6.2. Correspondence between puzzle pieces containing post-renormalizable critical points 6.3. Geometry of the puzzle pieces around other critical points 6.4. Proof of the Reduced Rigidity Theorem from rigidity in the infinitely renormalizable case The authors gratefully acknowledge support from the EPSRC GR R73171 01 and GR A11502 01 . WS is also supported by the Bai Ren Ji Hua project of the CAS. The authors would also like to thank the referee for his comments. 750 O. KOZLOVSKI W. SHEN AND S. VAN STRIEN 7. Rigidity in the infinitely renormalizable case assuming the Key Lemma 7.1. Properties of deep renormalizations 7.2. Compositions of real quadratic polynomials 7.3. Complex bounds 7.4. Puzzle geometry control 7.5. Gluing 8. Proof of the Key Lemma from upper and lower bounds 8.1. Construction of the enhanced nest 8.2. Properties of the enhanced nest 8.3. Proof of the Key Lemma assuming upper and lower bounds 9. Real bounds 10. Lower bounds for the enhanced nest 11. Upper bounds for the enhanced nest 11.1. Pulling-back domains along a chain 11.2. Proof of an
