The $C^1$ fixed point conjecture

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ mapping with $f(0)=0$ . Assume that every eigenvalue of $f'(x)$ has absolute value less than $1$ for every $x\in\mathbb{R}^n$ .

$C^1$ fixed point conjecture. The origin $0$ is the unique fixed point of $f$ .

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 1

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

The $C^1$ fixed point conjecture
Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map with $f(0)=0$ , and let $J(f)$ denote its Jacobian matrix. The $C^1$ fixed point conjecture. If the eigenvalues of $J(f)$ have absolute value less than $1$ at every point, then $0$ is the unique fixed point of $f$ . The paper states that, for polynomial maps, this conjecture is equivalent to the Jacobian Conjecture, while the general $C^1$ formulation remains open in the source.
source: L. Andrew Campbell, “Unipotent Jacobian Matrices and Univalent Maps”, arXiv:math/9907157 (1999).

Sources & referencesView supporting material

Primary source

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

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The $C^1$ fixed point conjecture

Equivalent formulations 1