Klasifikacija inverznih limit s poševnimi šotorskimi veznimi funkcijami

Črepnjak, Matevž

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Title:	Klasifikacija inverznih limit s poševnimi šotorskimi veznimi funkcijami
Authors:	ID Črepnjak, Matevž (Author) ID Banič, Iztok (Mentor) More about this mentor... ID Milutinović, Uroš (Comentor)
Files:	DR_Crepnjak_Matevz_2013.pdf (710,54 KB) MD5: 291B79C8978D980AC91E090F5BF7BFE0
Language:	Slovenian
Work type:	Dissertation
Typology:	2.08 - Doctoral Dissertation
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	V doktorski disertaciji bomo preučevali homeomorfnost inverznih limit inverznih zaporedij enotskih intervalov [0,1] s poševnimi šotorskimi veznimi funkcijami glede na lego vrhov poševnih šotorskih funkcij. Za poljubna $a,bin [0,1]$ je poševna šotorska funkcija $f_{(a,b)}:0,1]rightarrow [0,1]$ definirana kot večlična funkcija, katere graf $Gamma (f_{(a,b)})$ je unija daljic od $(0,0)$ do $(a,b)$ in od $(a,b)$ do $(1,0)$ . Točko $(a,b)$ imenujemo vrh poštevne šotorske funkcije $f_{(a,b)}$ . V prvem poglavju bomo predstavili inverzne limite inverznih zaporedij kompaktnih metričnih prostorov tako z enoličnimi kot večličnimi veznimi preslikavami. Predstavili bomo tudi Ingramovo domnevo, ki je glavna motivacija za preučevanje inverznih limit s poševnimi šotorskimi veznimi funkcijami. V drugem poglavju doktorske disertacije bomo govorili o inverznih limitah, ki so homeomorfne Brouwer-Janiszewski-Knasterjevemu kontinuumu. Natančneje, spoznali bomo nekatere primere inverznih limit inverznih zaporedij zaprtih enotskih intervalov $[0,1]$ s poševnimi šotorskimi veznimi preslikavami z vrhom v produktu $[0,1]times[0,1]$ , ki so homeomorfne Brouwer-Janiszewski-Knasterjevemu kontinuumu. V tretjem poglavju bomo govorili o klasifikaciji inverznih limit inverznih zaporedij zaprtih enotskih intervalov $[0,1]$ s poševnimi šotorskimi veznimi funkcijami z vrh-om v produktu $[0,1]times[0,1]$ . Izpeljali bomo pogoje za homeomorfnost posebnih pri-me-rov inverznih limit s poševnimi šotorskimi funkcijami. Posledično bomo videli, kdaj te inverzne limite niso homeomorfne. Tako bomo v produktu zaprtih intervalov $[0,1]times[0,1]$ predstavili takšne podmnožice, za katere bo veljalo naslednje: če vrhova poševnih šotorskih funkcij pripadata isti podmnožici, tedaj sta pripadajoči inverzni limiti homeomorfni, in če vrhova poševnih šotorskih funkcij pripadata različnim podmnožicam, tedaj pripadajoči inverzni limiti nista homeomorfni. Omenimo, da razdelitev $[0,1]times[0,1]$ na omenjene podmnožice ne bo popolna, saj se je problem klasifikacije takih inverznih izkazal kot zahteven in je postal zanimiv izziv mnogim raziskovalcem na tem področju. V četrtem poglavju bomo opisali še nekaj izvirnih rezultatov o hiperprostoru $2^{prod[0,1]}$ , opremljenim s Hausdorffovo metriko. Osredotočili se bomo na poti in loke, ki potekajo natanko skozi inverzne limite inverznih zaporedij enotskih zaprtih intervalov $[0,1]$ s poševnimi šotorskimi veznimi funkcijami z vrhom v produktu zaprtih enotskih intervalov $[0,1]times[0,1]$ . V zadnjem poglavju se bomo posvetili še odprtim problemom, ki se tičejo klasifikacije inverznih limit inverznih zaporedij zaprtih enotskih intervalov $[0,1]$ s po-šev-ni-mi šo-tor-ski-mi veznimi funkcijami z vrhovi v produktu $[0,1]times[0,1]$ . Opisali bomo tudi zanimive probleme, ki so nastali ob razvijanju disertacije in še niso rešeni. Prikazali bomo ideje in potencialne pristope za njihovo reševanje.
Keywords:	kontinuum, Brouwer-Janiszewski-Knasterjev kontinuum, inverzna limita, inverzno zaporedje, navzvgor polzvezna funkcija, večlična funkcija, vezna funkcija, šotorska funkcija, Ingramova domneva, s potmi povezan prostor, hiperprostor.
Place of publishing:	[Maribor
Publisher:	M. Črepnjak]
Year of publishing:	2013
PID:	20.500.12556/DKUM-40963
UDC:	515.126(043.3)
COBISS.SI-ID:	19970824
NUK URN:	URN:SI:UM:DK:8BPVQK2O
Publication date in DKUM:	08.07.2013
Views:	2923
Downloads:	294
Metadata:
Categories:	FNM
:	ČREPNJAK, Matevž, 2013, Klasifikacija inverznih limit s poševnimi šotorskimi veznimi funkcijami [online]. Doctoral dissertation. Maribor : M. Črepnjak. [Accessed 22 April 2025]. Retrieved from: https://dk.um.si/IzpisGradiva.php?lang=eng&id=40963 Copy citation

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Secondary language

Language:	English
Title:	Classifications of generalized tent maps inverse limits
Abstract:	In this dissertation we will study when the inverse limits of inverse sequences of unit intervals $[0,1]$ with generalized tent bonding function respecting on a position of top vertex of generalized tent functions are homeomorphic. For any $a, b in [0,1]$ , the generalized tent function $f_{(a,b)} : [0, 1] to [0, 1]$ is defined as the set-valued function with the graph $Gamma(f_{(a,b)})$ being the union of the segment from $(0, 0)$ to $(a, b)$ and the segment from $(a, b)$ to $(1, 0)$ . The point $(a, b)$ is called the top point of the generalized tent function $f_{(a,b)}$ . In the first chapter we will introduce the inverse limits of inverse sequences of compact metric spaces with both single and set-valued bonding functions. We present the Ingram conjecture, which is the main motivation for considering inverse limits with generalized tent bonding functions. In the second chapter of the dissertation we are dealing with inverse limits, homeomorphic to the Brouwer-Janiszewski-Kna-ster continuum. More precisely, we will give examples of the inverse limits of inverse sequences of unit intervals $[0,1]$ with generalized tent bonding functions with the top vertex in the product $[0,1]times[0,1]$ which are homeomorphic to the Brouwer-Janiszewski-Kna-ster continuum. In the third chapter we will deal with the classification of inverse limits of inverse sequences of unit intervals $[0,1]$ with generalized tent bonding functions with the top vertex in the product $[0,1]times[0,1]$ . In special cases we will prove the conditions when inverse limits with generalized tent functions are homeomorphic. Consequently we will see when these inverse limits are not homeomorphic. In the product of unit intervals $[0,1]times[0,1]$ we will describe such subsets, for which the following holds true: if a top vertices of generalized tent maps are in the same subset, then the corresponding inverse limits are homeomorphic, and if a top vertices of generalized tent maps are not in the same subset, then the corresponding inverse limits are not homeomorphic. Let us also mention that the dividing of $[0,1]times[0,1]$ of the above subsets will not be complete, since the problem of classifying such inverse limits is very complicated, becoming an interesting challenge to many researchers in the field. In the fourth chapter we will describe the original results about hyperspace $2^{prod[0,1]}$ equipped with the Hausdorff metric. We will concentrate on the paths and arcs that go only through inverse limits of inverse sequences of unit intervals $[0,1]$ with generalized tent bonding functions with top vertices in the product $[0,1]times[0,1]$ In the last chapter of dissertation we will focus on the open problems of classifications of inverse limits of inverse sequences of unit intervals $[0,1]$ with generalized tent bonding functions with top vertices in product $[0,1]times[0,1]$ . We will describe the interesting problems which occurred during the development of the thesis and which are not solved yet. Also we will show the ideas and potential approaches for their solving.
Keywords:	continuum, Brouwer-Janiszewski-Knaster continuum, inverse limit, inverse sequence, upper semi-continuous function, set-valued function, bonding function, tent map, Ingram conjecture, path-connected space, hyperspace.

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