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Added date when the conjecture was posed
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Martin Väth
Martin Väth
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The famous Nussbaum conjecture stated that every continuous map of a closed ball in a Banach space with a compact iterate (i.e. the iterate has relatively compact range) has a fixed point. Again Robert Cauty (see my previous post) proved it 2015 in the positive by showing that even a Lefschetz type fixed point theorem for maps with compact iterates holds:

  • Cauty, Robert, Un théorème de Lefschetz–Hopf pour les fonctions à itérées compactes, Crelle Journal für die reine und angewandte Mathematik 2017 (729), https://doi.org/10.1515/crelle-2014-0134

The conjecture was formulated in about 1970.

As Robert Nussbaum once pointed out, the attractivity of this conjecture lied in the fact that it is apparently so simple to prove, and that it can in fact be shown relatively easily under mild additional hypotheses (differentiability is such an “obviously” sufficient hypothesis, or that the map is even condensing, or that the range of some iterate has a locally nice topological structure, ...), but the longer one works on the problem, the harder it seems, and the less likely that one does not need any additional hypothesis. Many novelties in the field were inspired by proofs under such additional hypotheses.

The famous Nussbaum conjecture stated that every continuous map of a closed ball in a Banach space with a compact iterate (i.e. the iterate has relatively compact range) has a fixed point. Again Robert Cauty (see my previous post) proved it 2015 in the positive by showing that even a Lefschetz type fixed point theorem for maps with compact iterates holds:

  • Cauty, Robert, Un théorème de Lefschetz–Hopf pour les fonctions à itérées compactes, Crelle Journal für die reine und angewandte Mathematik 2017 (729), https://doi.org/10.1515/crelle-2014-0134

As Robert Nussbaum once pointed out, the attractivity of this conjecture lied in the fact that it is apparently so simple to prove, and that it can in fact be shown relatively easily under mild additional hypotheses (differentiability is such an “obviously” sufficient hypothesis, or that the map is even condensing, or that the range of some iterate has a locally nice topological structure, ...), but the longer one works on the problem, the harder it seems, and the less likely that one does not need any additional hypothesis. Many novelties in the field were inspired by proofs under such additional hypotheses.

The famous Nussbaum conjecture stated that every continuous map of a closed ball in a Banach space with a compact iterate (i.e. the iterate has relatively compact range) has a fixed point. Again Robert Cauty (see my previous post) proved it 2015 in the positive by showing that even a Lefschetz type fixed point theorem for maps with compact iterates holds:

  • Cauty, Robert, Un théorème de Lefschetz–Hopf pour les fonctions à itérées compactes, Crelle Journal für die reine und angewandte Mathematik 2017 (729), https://doi.org/10.1515/crelle-2014-0134

The conjecture was formulated in about 1970.

As Robert Nussbaum once pointed out, the attractivity of this conjecture lied in the fact that it is apparently so simple to prove, and that it can in fact be shown relatively easily under mild additional hypotheses (differentiability is such an “obviously” sufficient hypothesis, or that the map is even condensing, or that the range of some iterate has a locally nice topological structure, ...), but the longer one works on the problem, the harder it seems, and the less likely that one does not need any additional hypothesis. Many novelties in the field were inspired by proofs under such additional hypotheses.

Source Link
Martin Väth
Martin Väth
  • 1.9k
  • 1
  • 6
  • 12

The famous Nussbaum conjecture stated that every continuous map of a closed ball in a Banach space with a compact iterate (i.e. the iterate has relatively compact range) has a fixed point. Again Robert Cauty (see my previous post) proved it 2015 in the positive by showing that even a Lefschetz type fixed point theorem for maps with compact iterates holds:

  • Cauty, Robert, Un théorème de Lefschetz–Hopf pour les fonctions à itérées compactes, Crelle Journal für die reine und angewandte Mathematik 2017 (729), https://doi.org/10.1515/crelle-2014-0134

As Robert Nussbaum once pointed out, the attractivity of this conjecture lied in the fact that it is apparently so simple to prove, and that it can in fact be shown relatively easily under mild additional hypotheses (differentiability is such an “obviously” sufficient hypothesis, or that the map is even condensing, or that the range of some iterate has a locally nice topological structure, ...), but the longer one works on the problem, the harder it seems, and the less likely that one does not need any additional hypothesis. Many novelties in the field were inspired by proofs under such additional hypotheses.

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