On homeomorphic and diffeomorphic solutions of the Abel equation on the plane

Zbigniew Leśniak

Annales Polonici Mathematici (1993)

  • Volume: 58, Issue: 1, page 7-18
  • ISSN: 0066-2216

Abstract

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We consider the Abel equation φ[f(x)] = φ(x) + a on the plane ℝ², where f is a free mapping (i.e. f is an orientation preserving homeomorphism of the plane onto itself with no fixed points). We find all its homeomorphic and diffeomorphic solutions φ having positive Jacobian. Moreover, we give some conditions which are equivalent to f being conjugate to a translation.

How to cite

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Zbigniew Leśniak. "On homeomorphic and diffeomorphic solutions of the Abel equation on the plane." Annales Polonici Mathematici 58.1 (1993): 7-18. <http://eudml.org/doc/262502>.

@article{ZbigniewLeśniak1993,
abstract = { We consider the Abel equation φ[f(x)] = φ(x) + a on the plane ℝ², where f is a free mapping (i.e. f is an orientation preserving homeomorphism of the plane onto itself with no fixed points). We find all its homeomorphic and diffeomorphic solutions φ having positive Jacobian. Moreover, we give some conditions which are equivalent to f being conjugate to a translation. },
author = {Zbigniew Leśniak},
journal = {Annales Polonici Mathematici},
keywords = {functional Abel equation; free mapping; homeomorphic solutions; diffeomorphic solutions; Abel equation},
language = {eng},
number = {1},
pages = {7-18},
title = {On homeomorphic and diffeomorphic solutions of the Abel equation on the plane},
url = {http://eudml.org/doc/262502},
volume = {58},
year = {1993},
}

TY - JOUR
AU - Zbigniew Leśniak
TI - On homeomorphic and diffeomorphic solutions of the Abel equation on the plane
JO - Annales Polonici Mathematici
PY - 1993
VL - 58
IS - 1
SP - 7
EP - 18
AB - We consider the Abel equation φ[f(x)] = φ(x) + a on the plane ℝ², where f is a free mapping (i.e. f is an orientation preserving homeomorphism of the plane onto itself with no fixed points). We find all its homeomorphic and diffeomorphic solutions φ having positive Jacobian. Moreover, we give some conditions which are equivalent to f being conjugate to a translation.
LA - eng
KW - functional Abel equation; free mapping; homeomorphic solutions; diffeomorphic solutions; Abel equation
UR - http://eudml.org/doc/262502
ER -

References

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  1. [1] S. A. Andrea, On homeomorphisms of the plane which have no fixed points, Abh. Math. Sem. Hamburg 30 (1967), 61-74. Zbl0156.43704
  2. [2] S. A. Andrea, The plane is not compactly generated by a free mapping, Trans. Amer. Math. Soc. 151 (1970), 481-498. Zbl0205.54002
  3. [3] R. Engelking and K. Sieklucki, Topology. A Geometric Approach, Sigma Ser. Pure Math. 4, Heldermann, Berlin 1992. 
  4. [4] T. Homma and H. Terasaka, On the structure of the plane translation of Brouwer, Osaka Math. J. 5 (1953), 233-266. Zbl0051.14701
  5. [5] M. Kuczma, On the Schröder equation, Rozprawy Mat. 34 (1963). Zbl0121.33703
  6. [6] R. Sikorski, Advanced Calculus. Functions of Several Variables, Monograf. Mat. 52, PWN, Warszawa 1969. Zbl0182.37901
  7. [7] M. C. Zdun, On continuous iteration groups of fixed-point free mapping in ℝ² space, in: Proc. European Conference on Iteration Theory, Batschuns 1989, World Scientific, Singapore 1991, 362-368. 

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