Dynamics of typical Baire-1 functions on the interval

Abstract

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>I</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:math> {I=[0,1]} , and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>B</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:math> {bB_{1}} be the set of Baire-1 self-maps of I . For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>B</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> {f\in bB_{1}} , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>Λ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>f</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mo>⋃</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi>I</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mi>ω</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>f</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {\Lambda(f)=\bigcup_{x\in I}\omega(x,f)} be the set of ω-limit points of f . We prove the following: (i) There exists a residual subset S of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>B</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:math> {bB_{1}} such that for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mi>S</m:mi> </m:mrow> </m:math> {f\in S} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi>I</m:mi> </m:mrow> </m:math> {x\in I} the ω-limit set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ω</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>f</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\omega(x,f)} is contained in the set of points at which f is continuous, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ω</m:mi> <m:mo>⁢</m:mo> <m:mrow>