Publications of Michael Shub-February 2008
 

  1. Expanding Endomorphisms of Flat Manifold (with D. Epstein), Topology, Vol. 7, (1968), 139-141.
  2. Endomorphisms of Compact Differentiable Manifolds, Amer. J. Math. Vol. XCI, No. 1, (January 1969), 175-199.
  3. Periodic Orbits of Hyperbolic Diffeomorphisms and Flows, Bull.  AMS Vol. 1 (January 1969), 57-58.
  4. Future Stability is not Generic (with R.F. Williams), Proc. AMS Vol. 22 (1969), 483-484.
  5. Expanding Maps, Proc. Symposia in Pure Mathematics Vol. XIV (1970), 273-276.
  6. Neighborhoods of Hyperbolic Sets (with M. Hirsch, J. Palis, and C. Pugh), Inventiones Math. Vol. 9 (1970). 121-134.
  7. Linearization of Normally Hyperbolic Diffeomorphisms and Flows (with C. Pugh), Inventiones Math. Vol. 10, Fasc. 3 (1970), 187-198.
  8. Omega Stability Theorem for Flows (with C. Pugh), Inventiones Math. Vol. 11 (1970), 150-158.
  9. Invariant Manifolds (with M. Hirsch and C. Pugh), Bull. AMS, (September 1970), 1015-1019.
  10. Relative Equilibria and Diagonals, appendix to S. Smale's paper ``Problems on the Nature of Relative Equilibria in Celestial Mechanics", Manifolds, Proc. MUFFIC Summer School on Manifolds (1970), Amsterdam, Springer Lecture Notes in Mathematics Vol. 197, (1971), 199-201.
  11. Ergodic Elements of Ergodic Actions (with C. Pugh), Compositio Math. Vol. 23 (1971), 115-121.
  12. Instability, Symposium on Differential Equations and Dynamical Systems, Springer Lecture Notes in Mathematics Vol. 206 (1971), 28-29.
  13. Topologically Transitive Diffeomorphisms of T4, Symposium on Differential Equations and Dynamical Systems, Springer Lecture Notes in Mathematics Vol. 206, 39-40.
  14. Ergodicity of Anosov Actions (with C. Pugh), Inventiones Math.  Vol. 15, 1-23 (1972).
  15. Structurally Stable Diffeomorphisms are Dense, Bull. AMS, September 1972, 817-818.
  16. Beyond Hyperbolicity (with S. Smale), Annals of Mathematics, Vol. 96, No. 3 (Nov. 1972) 587-91.
  17. Morse-Smale Diffeomorphisms are Unipotent on Homology, in Dynamical Systems (Ed. M.M.Peixoto), Academic Press, New York, 1973, 489-492.
  18. Stability and Genericity for Diffeomorphisms,in Dynamical Systems, ibid., 493-514.
  19. Dynamical Systems, Filtrations and Entropy, Bull. Amer. Math. Soc.  Vol. 80 (1974), 27-41.
  20. Stability of Foliations (with H.I. Levine) Trans. of Amer. Math.  Soc. Vol. 184 (1973), 419-437.
  21. A Remark on the Lefschetz Fixed Point Formula for Differentiable Maps (with D. Sullivan), Topology, Vol. 13 (1974), 189-191.
  22. Homology Theory and Dynamical Systems (with Dennis Sullivan), Topology, Vol. 14 (1975), 109--132.
  23. Axiom A actions, (with C.C. Pugh), Inventiones Mathematicae, Vol. 29 (1974), 7-38.
  24. A Genericity Theorem in Topological Dynamics, (with J. Palis; C.C. Pugh, and D. Sullivan), in Dynamical Systems - Warwick 1974 (Ed. Manning) Springer-Verlag Lecture Notes in Math. #468, Berlin-Heidelberg-New York, 241-250.
  25. Topological Entropy and Stability in Dynamical Systems - Warwick 1974, ibid., 39-40.
  26. Homology Theory and Dynamical Systems, in Dynamical Systems - Warwick 1974, ibid., 36-38.
  27. On the Entropy Conjecture:  A Report on Conversations Among R. Bowen, M. Hirsch, A. Manning, C. Pugh, B. Sanderson, M. Shub, and R. Williams, recorded by Charles C. Pugh in Dynamical Systems- Warwick 1974, ibid., 257-261.
  28. Filtrations, Decompositions and Explosions, (with Z. Nitecki) Amer. J. of Math., vol. XCVII(1975), 1029-1048.
  29. Entropy and Stability, (with R. Williams), Topology, vol. 14, No. 4 (1975) 329-338.
  30. The Lefschetz Fixed Point Formula, Smoothness and Stability, in Dynamical Systems, an International Symposium, vol. 1, (Eds. Cesari et al.), Academic Press, 1976, 13-28.
  31. Invariant manifolds, (with M. Hirsch and C. Pugh), Springer-Lecture Notes on Mathematics, #583, Heidelberg 1977.
  32. Stability in Dynamical Systems in Colloques Internationaux Du Centre National de la Recherche Scientifique no. 259 ``Systemes Dynamiques Modeles Economiques", pp. 69-76.
  33. Alexander Cocycles for Dynamical Systems in Asterisque, Vol.(51), "Dynamical Systems III, Warsaw 1977", pp. 395-413, Societe Mathematique de France, Paris, 1978.
  34. On the Entropy of a Differentiable Map (with R. Sacksteder), Advances in Mathematics, Vol. 28 (1978), pp. 181-185.
  35. Entropy on Sphere Bundles (with R. Sacksteder), Advances in Mathematics, Vol. 28 (1978), pp. 174-177.
  36. Stabilit– Globale des Systemes Dynamiques, Asterisque, Vol. (56), 211 pages, Societe Mathematique de France, Paris, 1978.
  37. Some Dynamics of Pseudo-Anosov Diffeomorphisms (with A. Fathi), Expose 10, in Travaux de Thurston sur les Surfaces, Asterisque, Vol. 66-67, Societe Mathematiques de France, Paris 1979.
  38. Entropy, Linearity and Chain Recurrence (with D. Fried), Publ. Math. de l'.I.H.E.S. no. 50, 1979, pp. 451-462.
  39. Stable Manifolds for Maps (with D. Ruelle), in Global Theory of Dynamical Systems (Eds. Z. Nitecki and C. Robinson) Springer Lecture Notes in Mathematics #819, Springer-Verlag, Berlin, 1980, pp. 389-92.
  40. Differentiability and Continuity of Invariant Manifolds (with C.C. Pugh) in Nonlinear Dynamics (Ed. Robert H.G. Helleman) New York Academy of Sciences (1980), pp. 322-329.
  41. The Existence of Morse-Smale Diffeomorphisms (with J. Franks), Topology, Vol. 20 (1981), pp. 273-290.
  42. Suspending Subshifts (with C. Pugh), in Contributions to Geometry and Analysis (Eds. Clark, Percelli and Sacksteder) Johns Hopkins University Press (1981).
  43. On a Theory of Cost for Equations Solving (with S. Smale), announcement in Homotopy Methods and Global Convergence (Ed. B. Curtis Easves et al), Plenum Press, New York 1983, pp. 263-266.
  44. On the Average Cost of Solving Polynomial Equations, announcement, in Geometric Dynamics (Ed. J. Palis Jr.), Springer Lecture Notes in Mathematics  No. 1007, New York, 1983, pp. 719-724.
  45. Book Review of Geometric Theory of Dynamical Systems, by Palis and De Melo, The Mathematical Intelligencer, Vol. 6, No. 1, 1984, pp. 9-70.
  46. Computational Complexity:  On the Geometry of Polynomials and a theory of cost:  Part I (with S. Smale).  Annales Scientifiques de l'Ecole Normale Superieure, 4 Serie, t.18, 1985 pp. 107-142.
  47. Expanding Endomorphisms of the Circle Revisted (with D. Sullivan) Ergodic Theory and Dynamical Systems Vol. 5 (1985) pp. 285-289.
  48. The Integral Homology of Smale Diffeomorphisms (with M. Maller).  Topology, Vol. 24 (1985) pp. 153-164.
  49. On the Geometry of Polynomials and a Theory of Cost:  Part II (with S. Smale).  SIAM Journal on Computing, Vol. 15 (1986), pp. 145-161.
  50. On the Existence of Generally Convergent Algorithms (with S. Smale) Journal of Complexity, Vol.2 (1986) pp. 2-11.
  51. A Simple Unpredictable Pseudo - Random Number Generator (with L. Blum and M. Blum).  SIAM Journal on Computing, Vol. 15 (1986) pp. 364-383.
  52. Evaluating Rational Functions: Infinite Precision is Finite Cost and Tractable on Average.(with L. Blum).  SIAM Journal on Computing, Vol. 15 (1986) pp. 384-398.  Also Abstracted in IEEE 1984 FOCS.
  53. Some Remarks on Dynamical Systems and Numerical Analysis, in "Dynamical Systems and Partial Differential Equations:  Proceedings of the VII ELAM" L. Lara-Carrero and J. Lewowicz Editors, Equinoccio, Universidad Simon Bolivar, Caracas, 1986  pp. 69- 92.
  54. Average Condition Number for Solving Linear Equations (with N. Weiss, G. Wasilkowski and H. Wozniakowski)  Linear Algebra and its Applications, Vol. 83 (1986) pp. 79-102.
  55. Global Stability of Dynamical Systems, Springer, 1986 (this book is a translation of the French edition with added chapters and corrections).
  56. The Geometry and Topology of Dynamical Systems and Algorithms for Numerical Problems in Proceedings of the 1983 Beijing Symposium on Differential Geometry and Differential Equations (Editor Liao Shantao) Science Press, Beijing, China 1986 pp. 231-260.
  57. Some Linearly Induced Morse -Smale Systems, the QR - Algorithm and the Toda Lattice (with A. Vasquez) in Contemporary Mathematics, Vol. 64 The Legacy of Sonya Kovalevskaya, Linda Keen ed., American Mathematical Society, Providence 1987, pp. 181-194.
  58. Book Review of ``Information, Uncertainty, Complexity" by J.F. Traub G.W. Wasilkowski and H. Wozniakowski SIAM Review Vol. 29 (1987) pp. 495-7.
  59. On the Asymptotic Behavior of the Projective Rescaling Algorithm for Linear Programming, Journal of Complexity Vol. 3 (1987) pp. 258-269.
  60. The Newtonian Graph of a Complex Polynomial (with D. Tischler and R.F. Williams) SIAM Journal of Mathematical Analysis Vol. 19 (1988) pp. 246-256.
  61. Cr Stability of Periodic Solutions and Solution Schemes (with C.C. Pugh) Applied Math Letters Vol. 1 No. 3 (1988) pp. 281-285.
  62. Boundary Behaviour of Interior Point Algorithms in Linear Programming (with N. Meggido) Math of Operations Research Vol. 14 No.1 (1989) pp. 97-146.
  63. Ergodic Attractors (with C. Pugh) Trans. AMS. Vol. 312 (1989) pp. 1-54.
  64. On a Theory of Computation and Complexity over the Real Numbers; NP Completeness, Recursive Functions and Universal Machines (with L. Blum and S. Smale).  Bull. Amer. Math. Soc. (New Series) Vol. 21 (1989) pp. 1-46.  Also abstracted in the IEEE 1988 FOCS.
  65. On the Distance to the Zero Set of a Homogeneous Polynomial Journal of Complexity, Vol. 5 (1989) pp. 303-305.
  66. Book Review of Elements of differentiable dynamics and bifurcation theory by David Ruelle Bull. Amer. Math. Soc. Vol. 24 (1991) pp. 199-211.
  67. Can one always lower topological entropy? (with B. Weiss) Ergodic Theory and Dynamical Systems Vol. 11 (1991)  pp. 535-546.
  68. Unified Complexity Analysis for Newton LP Methods (with James Renegar) Math.  Programming  Vol. 53 (1992) pp. 1-16.
  69. Test Complexity of Generic Polynomials (with Peter Buergisser and Thomas Lickteig) Journal of Complexity Vol. 8 (1992) pp. 203-215.
  70. Complexity of Bezout's Theorem I: Geometrical Aspects (with Steve Smale) Journal of AMS Vol. 6 (1993), pp. 459-501.
  71. Complexity of Bezout's Theorem II: Volumes and Probabilities (with Steve Smale) in Computational Algebraic Geometry (F. Eyssette and A. Galligo, eds.) Progress in Mathematics Volume 109, Birkhauser, 1993, 267-285.
  72. Complexity of Bezout's Theorem III: Condition Number and Packing (with Steve Smale) Journal of Complexity Vol. 9 (1993), pp. 4-14.
  73. From Topology to Computation: Proceedings of the Smalefest, Editor (with M.W. Hirsch and J.E. Marsden), Springer, New York, 1993.
  74. On the Work of Steve Smale on the Theory of Computation ibid pp. 281-301.
  75. Some remarks on Bezout's Theorem and Complexity Theory ibid, pp. 443-455.
  76. How Many Eigenvalues of a Random Matrix are Real (with Alan Edelman and Eric Kostlan) Journal of AMS Vol. 7 (1994) pp. 247-267.
  77. Mysteries of Mathematics and Computation  Mathematical Intelligencer Vol. 16 No. 1 (1994) pp. 10-15.
  78. The Implicit Function Theorem Revisited IBM Journal of Research and Development Vol. 38 no. 3 (1994) pp. 259-264.
  79. Separation of Complexity Classes in Koiran's Weak Model (with F. Cucker and S. Smale) Theoretical Computer Science Vol. 133 (1994) pp. 3-14.
  80. Complexity and Bezout's Theorem V: Polynomial Time (with S. Smale) Theoretical Computer Science Vol. 133 (1994) pp. 141-164.
  81. Stably ergodic diffeomorhisms (with M. Grayson and C. Pugh) Annals of Math, Vol. 140 (1994) pp. 295-329.
  82. On the Intractibility of Hilbert's Nullstellensatz and an Algebraic Version of "NP not equal to P ? " (with S. Smale) Duke Math  J. Vol. 81 (1995) pp. 47-54.
  83. Complexity of Bezout's Theorem IV: Probability of Success; Extensions (with S. Smale)  SINUM Vol. 33 (1996) pp. 128-148.
  84. Complexity and Real Computation: A manifesto (with L. Blum, F. Cucker and Steve Smale) International Journal of Bifurcation and Chaos Vol. 6 (1996) pp. 3-26.
  85. Mathematics of Numerical Analysis (editor with J. Renegar and S.Smale) Lectures in Applied Math. Vol 32, American Math. Soc., Providence, R.I. 1996.
  86. Algebraic Settings for the Problem "P not equal to NP ? " (with L. Blum, F. Cucker and S. Smale) in "Mathematics of Numerical Analysis" op cit pp. 125-144.
  87. Stably Ergodic Skew Products (with R. Adler  and B. Kitchens) Discrete and Continuous Dynamical Systems Vol. 2 (1996) pp. 349-350.
  88. Generalized Knapsack Problems and Fixed Degree Separations (with F. Cucker) Theoretical Computer Science Vol. 161 (1996) pp. 301-306.
  89. Stable Ergodicity and Partial Hyperbolicity (with C.Pugh) in International Conference on      Dynamical Systems: Montevideo 1995, a tribute to Ricado Mane, Pitman Research Notes in Mathematics No. 362 (F. Ledrappier, J. Lewowicz and S. Newhouse eds.), (1997) 182-187.
  90. Foundations of Computational Mathematics (Ed. with F. Cucker) Springer 1997.
  91. Holder foliations (with C. Pugh and A. Wilkinson) Duke Math. J. Vol.86 (1997) 517-546. Correction Vol. 105 (2000) 105-106.
  92. Stably Ergodic Dynamical Systems and Partial Hyperbolicity (with C. Pugh) Journal of Complexity, Vol.13 (1997) pp 125-179.
  93. Stable Ergodicity in Homogeneous Spaces (with J. Brezin) Boletim Soc. Bras. Mat. Vol 28 (1997) 197-210.
  94. Stably Ergodicity and Stable Accessibility (with C. Pugh) in "Proceedings of US-Chinese Conference: Differential Equations and Applications held in Hangzhou, June 24-29, 1996" (eds. P.W. Bates et. al.) International Press, 258-267.
  95. Complexity and Real Computation, (with L.Blum, F. Cucker and S. Smale) Springer New York, 1998 432pp.
  96. Newton and Predictor Corrector Methods for Overdetermined Systems of Equations (with     J-P. Dedieu) IBM Research Report (available electronically at http://domino.watson.ibm.com/library/CyberDig.nsf/Home with keyword shub).
  97. The Distribution of the Maximum Condition Number on Great Circles Through a Fixed 2x2  Real Matrix  (with Debra Lewis).  Linear Algebra and Its Applications Vol. 297 (1999),  193-202.
  98. Pathological Foliations and Removable Zero Exponents (with Amie Wilkinson),  Inventventiones Mathematicae Vol.139 (2000), 495-508.
  99. Stable Ergodicity and Julienne Quasi-conformality (with C. Pugh).  Journal of the European  Math. Soc. Vol.2 (2000), 1-52.
  100. Multihomogeneous Newton Methods (with J-P. Dedieu), Math.of Comp. Vol. 69 (2000),  1071-1098.
  101. Newton's Method for Overdetermined Systems of Equations (with J-P. Dedieu),  Math of  Comp. Vol. 69, 1099-1115.
  102. Stably Ergodic Approximation: two examples (with Amie Wilkinson), Ergodic Theory  and Dynamical Systems Vol.20 (2000), 875-893.
  103. Deforming Factors in Idiopathic Scoliosis: A Mathematical Tool (with Joseph Margulies,  Roy Adler, Alan Kalvin, Marco Martens, Charles Tresser and Chai Wah Wu).  SPINE:State of the Art Reviews Vol 14 No.2 (2000),  Etiology of Adolescent Idiopathic Scoliosis: Current Trends and Relevance to New Treatment Approaches (Burwell,G. et. al., editors) Henley & Belfus, Philadelphia , 505-510
  104. On Simple Double Zeros and Badly Conditioned Zeros of Analytic Functions of n Variables (with J.P. Dedieu) , Math. of Comp. Vol.70 (2001), 319-327.
  105. Recent Results About Stable Ergodicity, (With K.Burns, C.Pugh and A.Wilkinson) Proceedings of Symposia in Pure Mathematics Vol 69-”Smooth Ergodic Theory and Its Applications (Katok, A., R de la Llave, Y. Pesin, H. Weiss, Eds) , AMS, Providence, R.I., 2001, 327-366
  106. From Dynamics to Computation and Back? in "Foundations of Computational Mathematics, Proceedings of the Smalefest 2000", F. Cucker and J.M. Rojas (Eds.) World Scientific, 2002, 423-432.
  107. Newton's Method on Manifolds and a Geometric Model for the Human Spine, (with Roy Adler, Jean-Pierre Dedieu, Joseph Margulies and Marco Martens) IMAJNA Vol.22 (2002) 359-390.
  108. Implicit Gamma Theorems (I): Pseudoroots and Pseudospectra (with J-P. Dedieu, M-H. Kim and F.Tisseur) Foundations of Computational Math. Vol.3 (2003) 1-33.
  109. On Random and Mean Exponents for Unitarily Invariant Probability Measures on $GL(n,C)$ (with J-P. Dedieu) in "Geometric Methods in Dynamical Systems (II)-Volume in Honor of Jacob Palis", Asterisque, Vol.287 (2003) 1-18 Soc. Math. De France.
  110. Random Versus Deterministic Exponents in a Rich Family of Diffeomorphisms (with F. Ledrappier, C. Simo and A. Wilkinson), Journal of Statistical Physics Vol.113 (2003), 85-149.
  111. Corrigendum to "Stable ergodicity and julienne quasi-conformality" ( J.Eur. Math. Soc. 2,1-52), (with C. Pugh and A. Starkov) J.Eur.Math.Soc. Vol 6 (2004), 149-151.
  112. Stable Ergodicity (with C.Pugh and an Appendix by A. Starkov) Bull. Amer. Math. Soc. Vol. 41 (2004), 1-41.
  113. Partial Differentiability of Invariant Splittings (with C. Pugh and A. Wilkinson) Journal of Statistical Physics Vol 114 (2004), 891-921.
  114. On the Curvature of the Central Path of Linear Programming Theory (with J.P. Dedieu and G. Malajovich), Foundations of Computational Mathematics Vol.5, (2005), 145 - 171
  115. What is a Horseshoe? Notices Amer. Math. Soc. Vol.52 (2005), 530-531.
  116. Newton Flow and Interior Point Methods in Linear Programming (with J.P. Dedieu) Internat. J. Bifur. Chaos Appl. Sci. Engrg. Vol. 15(2005),827--839.
  117. Convex Dynamics and Its Applications (with R. Adler, B. Kitchens, M. Martens, C.Pugh and C.Tresser) Ergodic Theory and Dynamical Systems Vol. 25 (2005), 321-352.
  118. Unique Ergodicity, Stable Ergodicity and the Mautner Phenomenon for Diffeomorhisms (with C.Pugh and A. Starkov) Discrete and Continuous Dynamical Systems, Vol.14 (2006), 845-855.
  119. Non-Zero Lyapunov Exponents Versus Mean Deterministic Exponents for a Twist Like Family of Diffeomorphisms of the Two Sphere- Extended Abstract in XIV International Congress on Mathematical Physics (J.C. Zambrini, ed.), World Scientific, 2006.
  120. All, Most, Some Differentiable Dynamical Systems, Proceedings of theInternational Congress of Mathematicians, Vol.3, Madrid, Spain,2006, European Mathematical Society , 99-120.
  121. Expanding maps of the circle rerevisited: Positive Lyapunov exponents in a rich family, (with E. Pujals and L. Robert) Ergodic Theory and Dynamical Sytems Vol 26 (2006), 1931-1937.
  122. Morse-Smale Systems, Scholarpedia, (2007)p.9494.
  123. Complexity of Bezout's Theorem VI: Geodesics in the Condition (Number) Metric, Foundations of Computational Mathematics , DOI 10.1007/s10208-007-9017-6, (2007).
  124. Complexity of Bezout's Theorem VII: Distance estimates in the condition metric (with Carlos Beltran) , Foundations of Computational Mathematics., DOI 10.1007/s10208-007-9018-5, (2007).
     

    Accepted for Publication

  125. Entropy estimates for a family of expanding maps of the circle (with R. de la LLave and C. Simo). To appear in an Aims Journal
  126. Dynamics of two dimensional Blaschke products (with Enrique Pujals). To appear Ergodic Theory and Dynamical Systems
     

    Submitted for Publication

  127. On the Geometry and Topology of the Solution Variety for Polynomial System Solving (with Carlos Beltran)
  128. Convexity properties of teh condition number (with Carlos Beltran, Jean-Pierre Dedieu, and Gregorio Malajovich)