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Keywords:
trajectory; first return continuity; $H$-connected function; ring of functions; D-ring; iteratively $H$-connected function
Summary:
In the paper the existing results concerning a special kind of trajectories and the theory of first return continuous functions connected with them are used to examine some algebraic properties of classes of functions. To that end we define a new class of functions (denoted $Conn^*$) contained between the families (widely described in literature) of Darboux Baire 1 functions (${\rm DB}_1$) and connectivity functions ($Conn$). The solutions to our problems are based, among other, on the suitable construction of the ring, which turned out to be in some senses an “optimal construction“. These considerations concern mainly real functions defined on $[0,1]$ but in the last chapter we also extend them to the case of real valued iteratively $H$-connected functions defined on topological spaces.
References:
[1] Bąkowska, A., Loranty, A., Pawlak, R. J.: On the topological entropy of continuous and almost continuous functions. Topology Appl. 158 (2011), 2022-2033. DOI 10.1016/j.topol.2011.06.049 | MR 2825356 | Zbl 1227.54022
[2] Biś, A., Nakayama, H., Walczak, P.: Modelling minimal foliated spaces with positive entropy. Hokkaido Math. J. 36 (2007), 283-310. DOI 10.14492/hokmj/1277472805 | MR 2347427 | Zbl 1137.57028
[3] Borsík, J.: Algebraic structures generated by real quasicontinuous functions. Tatra Mt. Math. Publ. 8 (1996), 175-184. MR 1475279 | Zbl 0914.54008
[4] Borsík, J.: Sums, differences, products and quotients of closed graph functions. Tatra Mt. Math. Publ. 24 (2002), 117-123. MR 1939288 | Zbl 1029.54021
[5] Bruckner, A. M.: Differentiation of Real Functions. Lecture Notes in Mathematics 659. Springer Berlin (1978). MR 0507448
[6] Čiklová, M.: Dynamical systems generated by functions with connected $G_{\delta}$ graphs. Real Anal. Exch. 30 (2004/2005), 617-638. MR 2177423
[7] Darji, U. B., Evans, M. J., Freiling, C., O'Malley, R. J.: Fine properties of Baire one functions. Fundam. Math. 155 (1998), 177-188. MR 1606523 | Zbl 0904.26003
[8] Darji, U. B., Evans, M. J., O'Malley, R. J.: A first return characterization for Baire one functions. Real Anal. Exch. 19 (1994), 510-515. MR 1282666 | Zbl 0840.26005
[9] Darji, U. B., Evans, M. J., O'Malley, R. J.: First return path systems: Differentiability, continuity, and orderings. Acta Math. Hung. 66 (1995), 83-103. DOI 10.1007/BF01874355 | MR 1313777 | Zbl 0821.26006
[10] Evans, M. J., O'Malley, R. J.: First-return limiting notions in real analysis. Real Anal. Exch. 29 (2003/2004), 503-530. MR 2083794
[11] Gibson, R. G., Natkaniec, T.: Darboux like functions. Real Anal. Exch. 22 (1996), 492-533. MR 1460971
[12] Grande, Z.: On the Darboux property of the sum of cliquish functions. Real Anal. Exch. 17 (1992), 571-576. MR 1171398 | Zbl 0762.26001
[13] Grande, Z.: On the sums and products of Darboux Baire*1 functions. Real Anal. Exch. 18 (1993), 237-240. MR 1205517
[14] Kellum, K. R.: Iterates of almost continuous functions and Sarkovskii's theorem. Real Anal. Exch. 14 (1989), 420-422. MR 0995981 | Zbl 0683.26004
[15] Korczak, E., Pawlak, R. J.: On some properties of essential Darboux rings of real functions defined on topological spaces. Real Anal. Exch. 30 (2004/2005), 495-506. MR 2177414
[16] Maliszewski, A.: Maximums of almost continuous functions. Real Anal. Exch. 30 (2004/2005), 813-818. MR 2177438
[17] Maliszewski, A.: Maximums of Darboux Baire one functions. Math. Slovaca 56 (2006), 427-431. MR 2267764 | Zbl 1141.26002
[18] Mikucka, A.: Graph quasi-continuity. Demonstr. Math. 36 (2003), 483-494. MR 1984357 | Zbl 1034.54010
[19] O'Malley, R. J.: First return path derivatives. Proc. Am. Math. Soc. 116 (1992), 73-77. DOI 10.2307/2159296 | MR 1097349 | Zbl 0762.26004
[20] Pawlak, R. J.: On some class of functions intermediate between the class $B_1^*$ and the family of continuous functions. Tatra Mt. Math. Publ. 19 (2000), 135-144. MR 1771030 | Zbl 0989.26002
[21] Pawlak, H., Pawlak, R. J.: First-return limiting notions and rings of Sharkovsky functions. Real Anal. Exch. 34 (2009), 549-563. MR 2569205 | Zbl 1183.26001
[22] Pawlak, R. J.: On the entropy of Darboux functions. Colloq. Math. 116 (2009), 227-241. DOI 10.4064/cm116-2-7 | MR 2520142 | Zbl 1232.37010
[23] Szuca, P.: Sharkovskiǐ's theorem holds for some discontinuous functions. Fundam. Math. 179 (2003), 27-41. DOI 10.4064/fm179-1-3 | MR 2028925 | Zbl 1070.26004
[24] Szuca, P.: Connected $G_{\delta}$ functions of arbitrarily high Borel class. Tatra Mt. Math. Publ. 35 (2007), 41-45. MR 2372433 | Zbl 1164.26006
[25] Vedenissoff, N.: Sur les fonctions continues dans des espaces topologiques. Fundam. Math. 27 (1936), 234-238 French. Zbl 0015.18005
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