Michael Shub | Academic Influence

Michael Shub's AcademicInfluence.com Rankings

Michael Shub

Mathematics

#1794

World Rank

#2869

Historical Rank

#731

USA Rank

Measure Theory

#1326

World Rank

#1671

Historical Rank

#471

USA Rank

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Mathematics

Why Is Michael Shub Influential?

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According to Wikipedia, Michael Ira Shub is an American mathematician who has done research into dynamical systems and the complexity of real number algorithms. Career 1967: Ph.D. and early career In 1967, Shub obtained his Ph.D. degree at the University of California, Berkeley with a thesis entitled Endomorphisms of Compact Differentiable Manifolds. In his Ph.D. thesis, he introduced the notion of expanding maps, which gave the first examples of structurally stable strange attractors. His advisor was Stephen Smale.

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Michael Shub's Published Works

Number of citations in a given year to any of this author's works

Total number of citations to an author for the works they published in a given year. This highlights publication of the most important work(s) by the author

Published Works

On a theory of computation and complexity over the real numbers: $NP$- completeness, recursive functions and universal machines (1989) (1144)
A Simple Unpredictable Pseudo-Random Number Generator (1986) (1016)
Global Stability of Dynamical Systems (1986) (821)
Endomorphisms of compact differentiable manifolds (1969) (323)
Complexity of Bézout’s theorem. I. Geometric aspects (1993) (305)
Newton's method on Riemannian manifolds and a geometric model for the human spine (2002) (285)
Complexity of Bezout's Theorem: III. Condition Number and Packing (1993) (277)
Complexity of Bezout’s Theorem II Volumes and Probabilities (1993) (268)
PERIODIC POINTS AND MEASURES FOR AXIOM A DIFFEOMORPHISMS (2010) (222)
Complexity of Bezout's theorem IV: probability of success; extensions (1996) (185)
On a theory of computation and complexity over the real numbers; np-completeness (1989) (185)
Pathological foliations and removable zero exponents (2000) (172)
Neighborhoods of hyperbolic sets (1970) (168)
Complexity of Bezout's Theorem V: Polynomial Time (1994) (164)
How many eigenvalues of a random matrix are real (1994) (154)
Boundary Behavior of Interior Point Algorithms in Linear Programming (1989) (152)
Stably Ergodic Dynamical Systems and Partial Hyperbolicity (1997) (149)
The local theory of normally hyperbolic, invariant, compact manifolds (1977) (141)
Expanding endomorphisms of the circle revisited (1985) (129)
Dynamical systems, filtrations and entropy (1974) (128)
A remark on the Lefschetz fixed point formula for differentiable maps (1974) (125)
Stably ergodic diffeomorphisms (1994) (120)
Ergodicity of Anosov actions (1972) (113)
Comparison of Two Pseudo-Random Number Generators (1982) (108)
Stable ergodicity and julienne quasi-conformality (2000) (107)
Homology theory and dynamical systems (1975) (103)
Newton's method for overdetermined systems of equations (2000) (94)
The Ω-stability theorem for flows (1970) (94)
Computational Complexity: On the Geometry of Polynomials and a Theory of Cost: II (1986) (90)
Some Remarks on Bezout’s Theorem and Complexity Theory (1993) (87)
On The Intractability Of Hilbert's Nullstellensatz And An Algebraic Version Of . . (1995) (87)
FILTRATIONS, DECOMPOSITIONS, AND EXPLOSIONS. (1975) (85)
Computational complexity: on the geometry of polynomials and a theory of cost. I (1985) (83)
Linearization of normally hyperbolic diffeomorphisms and flows (1970) (80)
Topologically transitive diffeomorphisms of T4 (1971) (79)
On a theory of computation over the real numbers; NP completeness, recursive functions and universal machines (1988) (73)
Complexity of Bezout’s Theorem VI: Geodesics in the Condition (Number) Metric (2007) (72)
Recent Results About Stable Ergodicity (2000) (63)
Ergodic elements of ergodic actions (1971) (60)
COMPLEXITY AND REAL COMPUTATION: A MANIFESTO (1996) (56)
On the Curvature of the Central Path of Linear Programming Theory (2003) (47)
Expanding endomorphisms of flat manifolds (1968) (46)
Structurally stable diffeomorphisms are dense (1972) (45)
Algebraic Settings for the Problem “P ≠ NP?” (1998) (42)
Complexity of Bezout’s Theorem VII: Distance Estimates in the Condition Metric (2009) (42)
Minimizing the discrete logarithmic energy on the sphere: The role of random polynomials (2011) (41)
Entropy and stability (1975) (38)
Partially Hyperbolic Dynamics, Laminations, and Teichmuller Flow (2007) (37)
Amino Acid Metabolism Conflicts with Protein Diversity (2014) (35)
Separation of Complexity Classes in Koiran's Weak Model (1994) (35)
Can one always lower topological entropy? (1991) (34)
Appendix to smale's paper: Diagonals and relative equilibria (1971) (33)
Multihomogeneous Newton methods (2000) (33)
Computing the Homology of Real Projective Sets (2016) (32)
The mathematics of numerical analysis (1996) (32)
Stable Ergodicity (2003) (31)
On the existence of generally convergent algorithms (1986) (31)
Genericity theorems in topological dynamics (1975) (30)
Morse-Smale Diffeomorphisms Are Unipotent on Homology (1973) (28)
A simple secure pseudo-random number generator (1982) (27)
Adaptative Step Size Selection for Homotopy Methods to Solve Polynomial Equations (2011) (27)
Unified complexity analysis for Newton LP methods (1992) (26)
On simple double zeros and badly conditioned zeros of analytic functions of n variables (2001) (25)
Stably ergodic approximation: two examples (2000) (25)
H\"older foliations, revisited (2011) (25)
The Newtonian graph of a complex polynomial (1988) (24)
Convexity Properties of the Condition Number (2008) (24)
Random Versus Deterministic Exponents in a Rich Family of Diffeomorphisms (2002) (23)
The mathematics of numerical analysis : 1995 AMS-SIAM Summer Seminar in Applied Mathematics, July 17-August 11, 1995, Park City, Utah (1996) (23)
Evaluating Rational Functions: Infinite Precision is Finite Cost and Tractable on Average (1984) (22)
All, most, some differentiable dynamical systems (2006) (22)
Entropy, linearity and chain-recurrence (1979) (22)
Test complexity of generic polynomials (1992) (21)
Stability and Genericity for Diffeomorphisms (1973) (19)
Convex dynamics and applications (2004) (18)
The existence of Morse–Smale diffeomorphisms (1981) (18)
Stable manifolds for maps (1980) (16)
On simple zeros of analytic functions of n variables (2001) (15)
Axiom A actions (1975) (15)
Periodic points on the $2$-sphere (2012) (14)
Condition Length and Complexity for the Solution of Polynomial Systems (2015) (14)
Stably ergodic skew products (1996) (14)
Average condition number for solving linear equations (1986) (13)
Corrigendum to: Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. 2, 1-52 (2004) (13)
Convexity Properties of the Condition Number II (2009) (13)
Newton Flow and interior Point Methods in Linear Programming (2005) (13)
Partial Differentiability of Invariant Splittings (2002) (12)
Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family (2006) (12)
From Topology to Computation: Proceedings of the Smalefest (2011) (11)
On the Geometry and Topology of the Solution Variety for Polynomial System Solving (2012) (11)
Mysteries of mathematics and computation (1994) (10)
A Note on the Finite Variance of the Averaging Function for Polynomial System Solving (2010) (10)
Entropy of a differentiable map (1978) (10)
Periodic orbits of hyperbolic diffeomorphisms and flows (1969) (9)
Cr Stability of periodic solutions and solution schemes (1988) (8)
DIFFERENTIABILITY AND CONTINUITY OF INVARIANT MANIFOLDS * (1980) (8)
The complexity and geometry of numerically solving polynomial systems (2012) (7)
On the asymptotic behavior of the projective rescaling algorithm for linear programming (1987) (7)
Dynamics of two-dimensional Blaschke products (2008) (7)
On the Work of Steve Smale on the Theory of Computation (1993) (7)
Generalized Knapsack Problems and Fixed Degree Separations (1996) (7)
Entropy estimates for a family of expanding maps of the circle (2008) (7)
Stable ergodicity in homogeneous spaces (1997) (6)
The integral homology of smale diffeomorphisms (1985) (6)
Future stability is not generic (1969) (5)
Smale’s Fundamental Theorem of Algebra Reconsidered (2012) (5)
Entropy on sphere bundles (1978) (5)
Entropy on sphere bundles (1978) (5)
Stability of foliations (1973) (5)
A Theory of NP-completeness and Ill-conditioning for Approximate Real Computations (2018) (4)
Morse-Smale systems (2007) (4)
Implicit Gamma Theorems (I): Pseudoroots and Pseudospectra (2003) (4)
Average polynomial time for eigenvector computations (2014) (3)
On the average cost of solving polynomial equations (1983) (3)
On the distance to the zero set of a homogeneous polynomial (1989) (3)
Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms (2006) (3)
Errata to "Stably ergodic skew products" (1999) (3)
Smale horseshoe (2007) (3)
Stable Manifolds for Hyperbolic Sets (1987) (3)
The Condition Number for Nonlinear Problems (1998) (2)
The implicit function theorem revisited (1994) (2)
Beyond Hyperbolicity (2019) (2)
A stably Bernoullian diffeomorphism that is not Anosov (1998) (2)
Stable and non-symmetric pitchfork bifurcations (2018) (2)
The Condition Number in ℙ(H(d)) (1998) (2)
Ornithine-containing lipid from Actinomyces globisporus (2004) (2)
Information, Uncertainty, Complexity (J. F. Traub, G. W. Wasilkowski and H. Woźniakowski) (1987) (2)
Topological entropy and stability (1975) (1)
Newton’s Method (1998) (1)
Foundations of computational mathematics : selected papers of a conference held at Rio de Janeiro, January 1997 (1997) (1)
Some Dynamics of Pseudo-Anosov Diffeomorphisms (2021) (1)
Foundations of Computational Mathematics (1997) (1)
Review: David Ruelle. Elements of differentiable dynamics and bifurcation theory (1991) (1)
Stably Ergodic (2007) (0)
Local product structure and local stability (1977) (0)
Ja n 20 05 Topological Classification of Holomorphic , Semi-Hyperbolic Germs , in “ Quasi-Absence ” of Resonances (2005) (0)
The distribution of the maximum condition number on great circles through a fixed 2×2 real matrix☆ (1999) (0)
Erratum to: Smale’s Fundamental Theorem of Algebra Reconsidered (2014) (0)
Editor's Foreword (1996) (0)
Forced smoothness of i: V → M (1977) (0)
The Cr section theorem and lipschitz jets (1977) (0)
Math and the Military (1987) (0)
Fundamental Theorem of Algebra: Complexity Aspects (1998) (0)
Computing the Homology of Real Projective Sets (2017) (0)
Transitive Transfer of Conndence: a Per- Fect Zero-knowledge Interactive Proto- Col for Sat and Beyond. 5.1 Additional Security for the User (1997) (0)
Decision Problems and Complexity over a Ring (1998) (0)
Deterministic Lower Bounds (1998) (0)
FROM DYNAMICS TO COMPUTATION AND BACK (2002) (0)
Non-zero random Lyapunov exponents versus mean deterministic exponents for a twist like family of diffeomorphisms of the two sphere (2006) (0)
Normally hyperbolic foliations and laminations (1977) (0)
1998 Best Paper Award (1999) (0)
Printed in the United Kingdom Convex dynamics and applications (2005) (0)
On a Theory of Cost for Equation Solving (1983) (0)
The Class NP and NP-Complete Problems (1998) (0)
Smale’s Fundamental Theorem of Algebra Reconsidered (2013) (0)
Complexity and the Condition Number (1998) (0)
Foreword (2005) (0)
Algebraic Settings for the Problem P 6 = Np?" 1. Statement of Main Theorems (1996) (0)
Some Separations of Complexity Classes (1998) (0)
Random and mean Lyapunov exponents for $\mathrm{GL}_n(\mathbb{R})$ (2022) (0)
The Lefschetz Fixed-Point Formula; Smoothness and Stability (1976) (0)
Disease Prediction with a Maximum Entropy Method (2021) (0)
PSEUDO-DIVISION MACHINE (I): A MODEL OF SYMBOLIC COMPUTATION (2007) (0)
Guest Editor's Introduction (1997) (0)
More Consequences of Hyperbolicity (1987) (0)
Preface (2003) (0)
Sequences of Filtrations (1987) (0)
Selected papers of a conference on Foundations of computational mathematics (1997) (0)
Editorial (2001) (0)
Computation over a Ring (1998) (0)
On the Geometry and Topology of the Solution Variety for Polynomial System Solving (2012) (0)
Nonuniform Complexity Classes (1998) (0)
Condition Numbers and the Loss of Precision of Linear Equations (1998) (0)
A Potpourri of Stability Results (1987) (0)
The linear theory of normal hyperbolicity (1977) (0)
Equivariant fibrations and nonwandering sets (1977) (0)
Definitions and First Properties of Computation (1998) (0)
Cusp Bifurcation in Metastatic Breast Cancer Cells (2022) (0)
Stable and non-symmetric pitchfork bifurcations In Memory of Professor (2020) (0)
Condition Length and Complexity for the Solution of Polynomial Systems (2016) (0)
Order out of chaos ? Some recent examples (0)
FORUM ON MILITARY FUNDING OF MATHEMATICS (1987) (0)
Pseudo hyperbolicity and plaque families (1977) (0)
Noncompactness and uniformity (1977) (0)
Real Computation and Complexity (Dagstuhl Seminar 04061) (2021) (0)
Erratum to: Smale’s Fundamental Theorem of Algebra Reconsidered (2014) (0)

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What Are Michael Shub's Academic Contributions?

Michael Shub is most known for their academic work in the field of mathematics. They are also known for their academic work in the fields of

Michael Shub has made the following academic contributions:

Complexity and Real Computation

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