On dynamics on the hyperspace of continua in dimension one

doctoral thesis

• Title (croatian): Dinamika na hiperprostoru jednodimenzionalnih kontinuuma
• Author: Domagoj Jelić
• Mentor: Piotr Oprocha https://orcid.org/0000-0002-0261-7229 (mentor)
• Mentor: Goran Erceg (komentor)
• Committee member: Nikola Koceić Bilan (predsjednik povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Committee member: Goran Erceg (član povjerenstva)
• Committee member: Ivančica Mirošević (član povjerenstva)
• Committee member: Piotr Oprocha https://orcid.org/0000-0002-0261-7229 (član povjerenstva)
• Committee member: Dino Peran (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2025-01-31, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Whenever we are given a selfmap $f$ of a compact metric space $X$, we can associate with it the induced mappings $\bar{f}$ and $\tilde{f}$ on the hyperspace $2^X$ of compact subsets of $X$ and the hyperspace $C(X)$ of continua in $X$, respectively, both defined in a natural way. This thesis concentrates on the relationship between the certain properties of selfmaps of some one-dimensional continua and of the induced mappings on their respective hyperspaces of subcontinua.... More Specifically, first we present the equality of the topological entropies of the arbitrary graph map and its induced mapping on the hyperspace of subgraphs. In order to do so, we characterize the set of subgraphs which are recurrent under the induced mapping. Further, we describe the structure of $\omega$-limit sets of the induced mapping on the hyperspace of continua for an arbitrary graph map. Strictly speaking, we prove that any subgraph is either asymptotically periodic, wandering or almost all of its iterates lie in a subsystem which is an almost 1-1 extension of irrational rotation, thus generalizing the earlier results obtained in the cases of compact intervals and topological trees. In the last chapter of the thesis, for every tree map $f:T \to T$, we establish the relation between the periods of the subtrees which are the periodic points of $\tilde{f}$ and the periods of periodic points of $f$ contained in those subtrees. We close the thesis by showing how the latter result can be used in order to prove some properties of the induced system on the hyperspace of continua. For any tree map $f:T \to T$, we provide complete description of the Birkhoff center of $(C(T),\tilde{f})$ and prove its almost equicontinuity. Less

Abstract (croatian)

Kad god je zadano preslikavanje $f$ na kompaktnom metričkom prostoru $X$, možemo mu pridružiti inducirana preslikavanja $\bar{f}$ i $\tilde{f}$ na hiperprostoru $2^X$ kompaktnih podskupova od $X$ i hiperprostoru $C(X)$ kontinuuma u $X$, redom, oba definirana na prirodan način. Kroz disertaciju smo usredotočeni na istraživanje odnosa između određenih svojstava preslikavanja na nekim jednodimenzionalnim kontinuumima i njihovih induciranih preslikavanja na pripadnim... More hiperprostorima potkontinuuma. Preciznije, prvo se bavimo uspostavljanjem jednakosti topoloških entropija proizvoljnih preslikavanja na topološkim grafovima i njihovih induciranih preslikavanja na hiperprostorima podgrafova. Kako bismo to postigli, karakteriziramo skup podgrafova koji su rekurentni s obzirom na inducirano preslikavanje. Nadalje, opisujemo strukutru $\omega$-graničnih skupova induciranog preslikavanja na hiperprostoru kontinuuma za proizvoljno preslikavanje topološkoga grafa. Strogo govoreći, pokazujemo da je svaki podgraf asimptotski periodičan, lutajuć ili su gotovo sve njegove iteracije sadržane u podstustavu koji je gotovo 1-1 proširenje iracionalne rotacije. Ovaj rezultat poopćava ranije dobivene rezultate u slučajevima kompaktnog intervala i topološkog stabla kao baznih prostora. U posljednjem poglavlju disertacije za svako preslikavanje $f:T \to T$ na topološkom stablu uspostavljamo odnos između perioda podstabala koja su periodičke točke od $\tilde{f}$ i perioda periodickih točaka of $f$ , sadržanih u tim podstablima. Za kraj pokazujemo kako upravo spomenuti rezultat može biti koristan pri dokazivanju određenih svojstava induciranog sustava na hiperprostoru kontinuuma. Naime, za proizvoljno preslikavanje $f:T \to T$ na topološkom stablu dajemo potpun opis Birkhoffovog centra od $(C(T),\tilde{f})$ i dokazujemo njegovu gotovo ekvikontinuiranost. Less

• Keywords: graph map; tree map; topological entropy; limit set; recurrence; hyperspace map; equicontinuous; Birkhoff center
• Keywords (croatian): preslikavanje grafa; preslikavanje stabla; topološka entropija; granični skup; rekurentnost; hiperprostorno preslikavanje; ekvikontinuiranost; Birkhoffov centar
• Language: english
• URN:NBN: urn:nbn:hr:217:105041
• Study programme: Title: Doctoral study; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica u području prirodnih znanosti (doktor/doktorica u području prirodnih znanosti)
• Type of resource: Text
• Extent: vi, 111 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-03-04 11:06:14