... The next year I was in Berkeley on a sabbatical, [Smale, 1981] was already written and Steve ... more ... The next year I was in Berkeley on a sabbatical, [Smale, 1981] was already written and Steve was teaching a course on it, which I took. Steve and I began collaborating during the semester; our work was finally published in [Shub-Smale, 1985] and [Shub-Smale, 1986a]. ...
In this paper we eliminate some of the most nettlesome hypotheses of our previous article, "Stabl... more In this paper we eliminate some of the most nettlesome hypotheses of our previous article, "Stably ergodic dynamical systems and partial hyperbolicity" , and expand the domain of stably ergodic, partially hyperbolic dynamical systems to include all partially hyperbolic affine diffeomorphisms of compact homogeneous spaces which have the accessibility property. Our main tool is a new concept -julienne quasi-conformality of the stable and unstable holonomy maps. An important feature of this is that the holonomy maps preserve all julienne density points.
Complexity theoretic aspects of continuation methods for the solution of square or underdetermine... more Complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors. In this paper we consider overdetermined systems where there are more equations than unknowns. We study Newton's method for such a system.
We study the geometry of the central paths of linear programming theory. These paths are the solu... more We study the geometry of the central paths of linear programming theory. These paths are the solution curves of the Newton vector field of the logarithmic barrier function. This vector field extends to the boundary of the polytope and we study the main properties of this extension: continuity, analyticity, singularities.
It has been conjectured that the stably ergodic diffeomorphisms are open and dense in the space o... more It has been conjectured that the stably ergodic diffeomorphisms are open and dense in the space of volume-preserving, partially hyperbolic diffeomorphisms of a compact manifold. In this paper we deal with two recalcitrant examples; the standard map cross Anosov and the ergodic automorphisms of the 4-torus. In both cases we show that they may be approximated by stably ergodic diffeomorphisms which have the stable accessibility property.
We study multihomogeneous analytic functions and a multihomogeneous Newton's method for finding t... more We study multihomogeneous analytic functions and a multihomogeneous Newton's method for finding their zeros. We give a convergence result for this iteration and we study two examples: the evaluation map and the generalized eigenvalue problem.
... The next year I was in Berkeley on a sabbatical, [Smale, 1981] was already written and Steve ... more ... The next year I was in Berkeley on a sabbatical, [Smale, 1981] was already written and Steve was teaching a course on it, which I took. Steve and I began collaborating during the semester; our work was finally published in [Shub-Smale, 1985] and [Shub-Smale, 1986a]. ...
In this paper we eliminate some of the most nettlesome hypotheses of our previous article, "Stabl... more In this paper we eliminate some of the most nettlesome hypotheses of our previous article, "Stably ergodic dynamical systems and partial hyperbolicity" , and expand the domain of stably ergodic, partially hyperbolic dynamical systems to include all partially hyperbolic affine diffeomorphisms of compact homogeneous spaces which have the accessibility property. Our main tool is a new concept -julienne quasi-conformality of the stable and unstable holonomy maps. An important feature of this is that the holonomy maps preserve all julienne density points.
Complexity theoretic aspects of continuation methods for the solution of square or underdetermine... more Complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors. In this paper we consider overdetermined systems where there are more equations than unknowns. We study Newton's method for such a system.
We study the geometry of the central paths of linear programming theory. These paths are the solu... more We study the geometry of the central paths of linear programming theory. These paths are the solution curves of the Newton vector field of the logarithmic barrier function. This vector field extends to the boundary of the polytope and we study the main properties of this extension: continuity, analyticity, singularities.
It has been conjectured that the stably ergodic diffeomorphisms are open and dense in the space o... more It has been conjectured that the stably ergodic diffeomorphisms are open and dense in the space of volume-preserving, partially hyperbolic diffeomorphisms of a compact manifold. In this paper we deal with two recalcitrant examples; the standard map cross Anosov and the ergodic automorphisms of the 4-torus. In both cases we show that they may be approximated by stably ergodic diffeomorphisms which have the stable accessibility property.
We study multihomogeneous analytic functions and a multihomogeneous Newton's method for finding t... more We study multihomogeneous analytic functions and a multihomogeneous Newton's method for finding their zeros. We give a convergence result for this iteration and we study two examples: the evaluation map and the generalized eigenvalue problem.
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Papers by Michael Shub