Contents Chapter 1. Introduction 1 1.1. Overview 3 1.2. Basic definitions and notations 5 1.3. Examples of non-smooth saddle-node bifurcations 6 1.4. The mechanism: Exponential evolution of peaks 15 Chapter 2. Statement of the main results and applications 21 2.1. A general setting for saddle-node bifurcations in qpf interval maps 21 2.2. Sink-source-orbits and the existence of SNA 23 2.3. Non-smooth bifurcations 26 2.4. Application to the parameter families 29 Chapter 3. Saddle-node bifurcations and sink-source-orbits 36 3.1. Equivalence classes of invariant graphs and the essential closure 36 3.2. Saddle-node bifurcations: Proof of Theorem 2.1 37 3.3. Sink-source-orbits and SNA: Proof of Theorem 2.4 42 Chapter 4. The strategy for the construction of the sink-source-orbits 44 4.1. The first stage of the construction 44 4.2. Dealing with the first close return 46 4.3. Admissible and regular times 50 4.4. Outline of the further strategy 51 Chapter 5. Tools for the construction 54 5.1. Comparing orbits 54 5.2. Approximating sets 59 5.3. Exceptional intervals and admissible times 62 5.4. Regular times 68 Chapter 6. Construction of the sink-source orbits: One-sided forcing 73 6.1. Proof of the induction scheme 77 Chapter 7. Construction of the sink-source-orbits: Symmetric forcing 92 7.1. Proof of the induction scheme 95 Bibliography 105 v

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