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The C1C^1 fixed point conjecture

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Let f:Rn→Rnf:\mathbb{R}^n\to\mathbb{R}^n be a C1C^1 mapping with f(0)=0f(0)=0. Assume that every eigenvalue of f′(x)f'(x) has absolute value less than 11 for every x∈Rnx\in\mathbb{R}^n.

C1C^1 fixed point conjecture. The origin 00 is the unique fixed point of ff.

The conjecture is discussed in connection with injectivity questions for mappings with unipotent Jacobian matrices, the Jacobian Conjecture, Chamberland's conjecture, and the Markus--Yamabe problem. The source presents related problems as still open.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The C1C^1 fixed point conjecture

    Let f:Rn→Rnf:\mathbb{R}^n\to\mathbb{R}^n be a C1C^1 map with f(0)=0f(0)=0, and let J(f)J(f) denote its Jacobian matrix. The C1C^1 fixed point conjecture. If the eigenvalues of J(f)J(f) have absolute value less than 11 at every point, then 00 is the unique fixed point of ff. The paper states that, for polynomial maps, this conjecture is equivalent to the Jacobian Conjecture, while the general C1C^1 formulation remains open in the source.

    source: L. Andrew Campbell, “Unipotent Jacobian Matrices and Univalent Maps”, arXiv:math/9907157 (1999).

References

Primary source

Victor Alexandrov, “Around Efimov's differential test for homeomorphism”, arXiv:2006.15322 (2020).

dynamical-systemsfixed-pointsreal-analysis
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