| 1. | Lyubich published in 1999 the first not computer assisted proof which also establishes hyperbolicity.
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| 2. | In particular, it would imply the important " hyperbolicity conjecture " mentioned above.
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| 3. | The first condition of the theorem is known as "'global hyperbolicity " '.
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| 4. | Later work by various authors gave alternate proofs of this fact and better information on the hyperbolicity.
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| 5. | Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.
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| 6. | It is reasonable to believe that any physical spacetime will satisfy the strongest causality condition : global hyperbolicity.
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| 7. | The hyperbolicity is essential to verify the picture discovered numerically by Feigenbaum and independently by Coullet and Tresser.
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| 8. | In particular he solved or made progress on several conjectures about the hyperbolicity of subvarieties of algebraic varieties.
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| 9. | For example, the following hyperbolicity results have led to new phenomena being discovered for the groups acting on them.
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| 10. | Campanino and Epstein also gave a proof of the fixed point without computer assistance but did not establish its hyperbolicity.
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