SILESIAN UNIVERSITY IN OPAVA Mathematical Institute in Opava Abstract of the Ph.D. Thesis Various types of chaos and entropy in dynamical systems Author: Supervisor: Mgr. Zuzana Roth doc. RNDr. ZdenˇekKoˇcan, Ph.D. May 2019, Opava Mathematical Analysis SLEZSKA´ UNIVERZITA V OPAVEˇ Matematick´y´ustav v Opavˇe Autorefer´at dizertaˇcn´ıpr´ace R˚uzn´etypy chaosu a entropie v dynamick´ych syst´emech Autor: Skolitel:ˇ Mgr. Zuzana Roth doc. RNDr. ZdenˇekKoˇcan, Ph.D. Duben 2019, Opava Matematick´aanal´yza V´ysledky uveden´ev t´eto dizertaˇcn´ıpr´aci byly z´ısk´any v pr˚ubˇehu m´eho doktorsk´eho studia oboru Matematick´a anal´yza na Matematick´em ´ustavu Slezsk´e univerzity v Opavˇev letech 2013 { 2019. V´yzkum byl podporov´an granty: SGS/2/2013, SGS/16/2016, SGS/18/2016 a SGS/18/2019. Dizertant: Mgr. Zuzana Roth Matematick´y´ustav v Opavˇe,Slezsk´auniverzita v Opavˇe Skolitel:ˇ doc. RNDr. ZdenˇekKoˇcan, Ph.D. Matematick´y´ustav v Opavˇe,Slezsk´auniverzita v Opavˇe Skol´ıc´ıpracoviˇstˇe:ˇ Matematick´y´ustav v Opavˇe, Slezsk´auniverzita v Opavˇe Oponenti: doc. RNDr. Marta Stef´ankov´a,ˇ Ph.D. Matematick´y´ustav v Opavˇe,Slezsk´auniverzita v Opavˇe doc. RNDr. Roman Hric, PhD. Fakulta pr´ırodn´ych vied, Univerzita Mateja Bela, Bansk´aBystrica Autorefer´at t´eto dizertaˇcn´ıpr´ace byl zveˇrejnˇendne 12. dubna 2019. St´atn´ıdoktorsk´azkouˇska a obhajoba dizertaˇcn´ıpr´ace se budou konat 05.06.2019 v 11.00 hod pˇred oborovu komis´ı doktorsk´eho studia - Matematick´a anal´yza, v zasedac´ım´ıstnosti rektor´atu Slezsk´euniverzity v Opavˇe. Dizertaˇcn´ıpr´ace je dostupn´av knihovnˇeMatematick´eho ´ustavu, Na Rybn´ıˇcku 626/1, 74601, Opava. Pˇredseda oborov´erady: prof. RNDr. Miroslav Engliˇs,DrSc. Matematick´y´ustav v Opavˇe Slezsk´auniverzita v Opavˇe Na Rybn´ıˇcku 626/1 74601 Opava Acknowledgments I sincerely thank my supervisor, ZdenˇekKoˇcan, as well as my coauthors, Jana Hant´akov´aand Ana Rodrigues, for their invaluable help and support. I am also very grateful for my coauthor, colleague, language corrector, ... ... (and last but not least) my husband, Samuel J. Roth. Table of Contents 1 Introduction .................................. 1 2 Motivation for the research ........................ 2 3 Terminology and notation ......................... 4 4 Main Results ................................. 9 [A] On the weakest version of distributional chaos. 9 [B] Distributional chaos and dendrites. 9 [C] Inequalities for entropy, Hausdorff dimension, and Lipschitz constants. 11 [D] Fair measures for countable-to-one maps. 12 5 Open problems ................................ 14 6 Publications .................................. 16 7 Citations by other authors ......................... 16 8 Conferences and other presentations .................. 17 9 References ................................... 18 1 Introduction Over the history of mankind we have always tried to determine what can be considered as order and what is already chaos. And so nowadays we have dozens of different definitions and levels of chaotic behaviour. It is not in the scope of this thesis to even name all of them (we will closely look at 3 different areas). The history of chaos theory in mathematics could go as far back as Lorenz (butterfly effect, 1960s), Poincar´e(sensitivity to initial conditions, n-body problem, 19th century), or even Kepler (planetary motion, 17th century)... but as Oestreicher said in [Oes07] \Lorenz had rediscovered the chaotic behavior of a nonlinear system, that of the weather, but the term chaos theory was only later given to the phenomenon by the mathematician James A. Yorke, in 1975." It is true that Yorke, together with Li, gave one of the first definitions of chaos in mathematics in [LY75], even though more commonly we hear Devaney's description used as the definition of chaos [Dev89]. The term entropy has been used in physics to describe chaotic behavior since approximately the 19th century (see [Cla56, Cla67]), but the popularity of this term came with information science in the mid 20th century [SWB51] while in math the attention was brought to entropy by Kolmogorov and Sinai in [Kol58, Sin59]. The term entropy can now be found in many areas of science; usually it means disorder or chaos. In mathematics a system with positive entropy can also be called h-chaotic. We can try to dig deeper in history for other hints of chaos definitions, but if we look at it closely, chaos theory and dynamical systems are relatively young branches of mathematics with good potential. For some more details about the history of different types of chaos and entropy see [Dow07, Oes07, CAM+05] and [Wal00, chapters: 4, 7, 8]. Out of the many different definitions of chaos and entropy, this thesis will mainly consider distributional chaos as defined in [SS94, BSS05]ˇ (and also its relation with Li-Yorke chaos [LY75]). For entropy we will focus on topological entropy [AKM65] and a special case of measure-theoretic entropy - \fair entropy" originally defined in [MR18]. 1 The thesis is based on 4 papers [A, B, C, D]. The first 2 are focused more on distributional chaos, while the other 2 deal with entropy. The main topic of this thesis involves the problem of determining the relationships between the different types of chaos, as well as the \stability" of each type. Stability can be understood as the persistence of chaos under: conjugacy, composition, extension, small pertubations, etc. The topic of stability also raises the question of topological invariants, which are not always easy to find (2 of my 4 articles introduce 2 new invariants [A, C]). The thesis is organized as follows. In the next section we give motivational comments and relations about the main types of chaos for this thesis. In section 3 we properly introduce the necessary notations, background and definitions. Section 4 points out our main results from all 4 articles. Open questions follow in section 5. Then we close with a list of publications, citations and presentations. In the appendices the reader can find full copies of the articles [A, B, C, D] along with statements confirming my coauthorship and the acceptance of article [C] for publication. 2 Motivation for the research If we look at continuous functions acting on the closed interval, a lot of the definitions of chaos coincide. Li and Yorke claimed that period 3 implies chaos [LY75], while Sm´ıtal (and Misiurewicz) showed that period 3 is actually not necessary for Li-Yorke (LY) chaos [MS88, Sm´ı86] already on the interval (even for C1 maps). However their examples also have topological entropy equal to 0. That led to an assumption that positive topological entropy is a stronger type of chaos than LY. When later Schweizer and Sm´ıtal defined distributional chaos [SS94] (abbreviated DC, or DC1, since later there were defined DC1, DC2, DC3 as subtypes of DC - for details see section 3), they also discoverd that the existence of a single DC-pair is equivalent to positive topological entropy for continuous maps on a closed interval. Later it also was shown that the existence of any kind of DC-pair is equivalent to the existence of an uncountabele DC1-set and positive topological entropy for continuous maps on 2 closed intervals and graphs, even though in general these concepts are distinguishable (see eg. [BSS05,ˇ Pik07, M´al07, KKKM11, Koˇc12]). It is also known that topological entropy is a topological invariant, and the same was shown about DC1 and DC2 (see [Wal00, SS04]).ˇ And so the question naturally arose how much \weaker" is the definition of DC3 in general spaces than the first 2 types? Well, it turned out that even though on the closed interval DC3 is equivalent to positive topological entropy, in general it is not even as strong as LY, moreover the property, as it is defined, is in some sense unstable [A]. Nevertheless, we had hoped to carry at least some of the results from intervals and graphs to dendrites, since they are often understood as a collection of intervals glued together, and so they are a natural next step from the interval and graphs to slightly more complicated spaces. The results of that investigation are found in [B]. On the other hand, a common task when studying entropy is to calculate its value, or at least find some upper or lower bounds. (Lower bounds are best when we want to show that the entropy is positive, and upper when we want to show that it is zero.) How can we connect these entropy bounds with the dimension, the metric, or other properties of the system? In the process of dealing with these questions, we were able to improve some older results from [DZG98], and as a bonus we found a new topological invariant [C]. Fair measures and fair entropy are the newest concepts we worked with. Both are introduced in [MR18] in an attempt to find (again) an easier way to calculate topological entropy. But instead of producing a new formula for topological entropy they lead instead to a lower bound (hfair ≤ htop) and a new topological invariant. We looked at the challenge of generalizing this concept; we managed to go beyond compact spaces and continuous mappings [D]. Moreover, just as Birkhoff's ergodic theorem describes forward trajectories, fair measures give us a tool for studying backward trajectories. 3 3 Terminology and notation If not indicated otherwise, we will use the following notation throughout the thesis. The pair (X; d) will be a non-empty compact metric (or at least metrizable) space with metric d. By f : X ! X we denote a continuous map, and f n denotes the nth 0 n+1 n iterate of f, for n 2 N0, so that f (x) = id(x) = x, f = f ◦ f . Dendrites are locally connected continua (nonempty, compact, connected metric spaces) not containing any simple closed curves.