Real Analysis Exchange

Loops of intervals and Darboux Baire 1 fixed point problem.

Piotr Szuca

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WWe show that the existence of a loop of covering intervals for connected $G_\delta$ real functions implies the existence of a periodic point with its period equal to the length of the cycle. As a corollary we get that the composition of $N$ connected $G_\delta$ real functions from a compact interval into itself has a fixed point for every natural number $N$. In particular, the composition of finitely many derivatives from $[0,1]$ to $[0,1]$ has a fixed point. This solve the a problem from [3].

Article information

Real Anal. Exchange, Volume 29, Number 1 (2003), 205-209.

First available in Project Euclid: 9 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}
Secondary: 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25] 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05] 54C30: Real-valued functions [See also 26-XX] 37E99: None of the above, but in this section

Derivatives connectivity functions Darboux functions Baire~1 functions Borel measurable functions composition loop of intervals $f$-cover


Szuca, Piotr. Loops of intervals and Darboux Baire 1 fixed point problem. Real Anal. Exchange 29 (2003), no. 1, 205--209.

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