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2003-2004 Loops of intervals and Darboux Baire 1 fixed point problem.
Piotr Szuca
Author Affiliations +
Real Anal. Exchange 29(1): 205-209 (2003-2004).


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].


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Piotr Szuca. "Loops of intervals and Darboux Baire 1 fixed point problem.." Real Anal. Exchange 29 (1) 205 - 209, 2003-2004.


Published: 2003-2004
First available in Project Euclid: 9 June 2006

zbMATH: 1065.26004
MathSciNet: MR2061304

Primary: 26A15
Secondary: 26A18 , 26A21 , 37E99 , ‎54C30

Keywords: $f$-cover , Baire~1 functions , Borel measurable functions , composition , connectivity functions , Darboux functions , derivatives , loop of intervals

Rights: Copyright © 2003 Michigan State University Press

Vol.29 • No. 1 • 2003-2004
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