Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences

We consider infinite sequences of superstable orbits (cascades) generated by systematic substitutions of letters in the symbolic dynamics of one-dimensional nonlinear systems in the logistic map universality class. We identify the conditions under which the topological entropy of successive words converges as a double exponential onto the accumulation point, and find the convergence rates analytically for selected cascades. Numerical tests of the convergence of the control parameter reveal a tendency to quantitatively universal double-exponential convergence. Taking a specific physical example, we consider cascades of stable orbits described by symbolic sequences with the symmetries of quasilattices. We show that all quasilattices can be realised as stable trajectories in nonlinear dynamical systems, extending previous results in which two were identified.Comment: This version: updated figures and added discussion of generalised time quasilattices. 17 pages, 4 figure

On the Lebesgue measure of Li-Yorke pairs for interval maps

We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for $f \times f$. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where $f$ has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.Comment: 41 pages, 3 figure

Wild attractors and thermodynamic formalism

Fibonacci unimodal maps can have a wild Cantor attractor, and hence be Lebesgue dissipative, depending on the order of the critical point. We present a one-parameter family $f_\lambda$ of countably piecewise linear unimodal Fibonacci maps in order to study the thermodynamic formalism of dynamics where dissipativity of Lebesgue (and conformal) measure is responsible for phase transitions. We show that for the potential $\phi_t = -t\log|f'_\lambda|$, there is a unique phase transition at some $t_1 \le 1$, and the pressure $P(\phi_t)$ is analytic (with unique equilibrium state) elsewhere. The pressure is majorised by a non-analytic $C^\infty$ curve (with all derivatives equal to 0 at $t_1 < 1$) at the emergence of a wild attractor, whereas the phase transition at $t_1 = 1$ can be of any finite order for those $\lambda$ for which $f_\lambda$ is Lebesgue conservative. We also obtain results on the existence of conformal measures and equilibrium states, as well as the hyperbolic dimension and the dimension of the basin of $\omega(c)$

Extremal properties of (epi)Sturmian sequences and distribution modulo 1

Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in distribution of real numbers modulo 1 via combinatorics on words, we survey some combinatorial properties of (epi)Sturmian sequences and distribution modulo 1 in connection to their work. In particular we focus on extremal properties of (epi)Sturmian sequences, some of which have been rediscovered several times

Dynamics of continued fractions and kneading sequences of unimodal maps

In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the alpha-continued fraction transformations T_alpha and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers.Comment: 21 pages, 3 figures. New section added with additional results and applications. Figures and references added. Introduction rearrange

Dynamical Systems and Matching Symmetry in beta-Expansions

Symbolic dynamics, and in particular β-expansions, are a ubiquitous tool in studying more complicated dynamical systems. Applications include number theory, fractals, information theory, and data storage. In this thesis we will explore the basics of dynamical systems with a special focus on topological dynamics. We then examine symbolic dynamics and β-transformations through the lens of sequence spaces. We discuss observations from recent literature about how matching (the property that the itinerary of 0 and 1 coincide after some number of iterations) is linked to when Tβ,⍺ generates a subshift of finite type. We prove the set of ⍺ in the parameter space for which Tβ,⍺ exhibits matching is symmetric and analyze some examples where the symmetry is both apparent and useful in finding a dense set of ⍺ for which Tβ,⍺ generates a subshift of finite type

Inverzni limesi preslikavanja na intervalu

This thesis studies topological properties of unimodal inverse limit spaces and planar embeddings of chainable continua in general. In the first part we study global and local properties of inverse limit spaces on the unit interval with a single bonding map coming from the tent family. We give symbolic description of arc-components and study the inhomogeneity points of the space. Specifically, we prove that the set of folding points is equal to the set of endpoints if and only if the critical orbit is persistently recurrent, answering the question of Alvin and Brucks from 2010. Also, we make a topological distinction of the arc-component containing the orientation reversing fixed point in the case when the critical orbit is non-recurrent which enables us to prove the Core Ingram conjecture in this case in the positive. To be more precise, we show that the cores of tent inverse limits for which the critical point is non-recurrent are non-homeomorphic for different slopes. The second part of the thesis studies non-equivalent planar embeddings of general chainable continua. We show that every chainable continuum which contains an indecomposable subcontinuum can be embedded in the plane in uncountably many strongly non-equivalent ways, and answer the question of Mayer from 1982 in the unimodal inverse limit case. We also study accessible sets of points of planar embeddings of chainable continua and give a positive answer to the question of Nadler and Quinn from 1972 in case when points are not contained in zigzags of bonding maps.Inverzni limesi daju efikasnu metodu za opis prostora dobivenih kao presjek ugnježđenog niza skupova u pripadajućem metričkom prostoru i kao takvi nalaze primjenu u raznim matematičkim područjima. Istaknimo na primjer primjenu inverznih limesa u opisu hiperboličkih atraktora. U tezi proučavamo lančaste kontinuume, odnosno inverzne limese u kojima su vezne funkcije preslikavanja na intervalima. U prvom dijelu se restriktiramo na unimodalne inverzne limese i proučamo njihova topološka svojstva. Takvi kontinuumi se često pojavljuju kao modeli čudnih atraktora ravninskih homeomorfizama. U drugom dijelu dajemo metodu za konstrukciju različitih planarnih smještenja lančastih kontinuuma koristeći metodu permutacije grafova veznih preslikavanja. Preslikavanje na intervalu koje fiksira krajnje točke i ima jedinstveni ekstrem u interioru intervala zovemo unimodalno. U tezi se restriktiramo na šatorske funkcije, koje, unatoč tome što su vrlo specifične, gotovo u potpunosti opisuju dinamička i topološka svojstva od interesa. Zanimaju nas lokalna i globalna topološka svojstva šatorskih inverznih limesa. Lokalno nas zanimaju točke koje nemaju (otvorene) okoline homeomorfne Kantorovom skupu (otvorenih) lukova. Takve točke zovemo točke nabiranja. U tezi dajemo karakterizaciju točaka nabiranja u terminima dinamičkih svojstava veznog preslikavanja i, specijalno, podskupa čije elemente zovemo krajnje točke. To su točke $x$ za koje vrijedi da ako su $A, B$ podkontinuumi koji sadrže $x$, onda $A \subset B$ ili $B \subset A$. Dajemo odgovor na pitanje koje su 2010 postavili Alvin i Brucks, odnosno pokazujemo sljedeći teorem. Teorem 1.1. Svaka točka nabiranja je krajnja točka ako i samo ako je kritična točka veznog preslikavanja uporno rekurentna. Naglasimo kako se uporna rekurentnost pojavila kao nužan uvjet za postojanje divljih atraktora unimodalnih preslikavanja na intervalu. Pod globalna topološka svojstva podrazumijevamo strukturu podkontinuuma, kompozanti i lučnih komponenti. U ovom smjeru postoji još mnogo otvorenih pitanja. U tezi dajemo simboličku karakterizaciju lučnih komponenti koristeći svojstva posebnog tipa krajnjih točaka koje zovemo spiralne točke. Svojstva specijalne lučne komponente, koja sadrži fiksnu točku jezgre i postoji u svakoj jezgri šatorskih inverznih limesa, nam omogućavaju da damo potpunu karakterizaciju jezgara u slučaju kada je kritična orbita nerekurentna. Dokazujemo sljedeće teoreme. Teorem 1.3. Ako je $1 < s < \tilde{s} < 2$ i kritične orbite odgovarajućih šatorskih preslikavanja $T_s$ i $T_{\tilde{s}}$ su beskonačne i nerekurentne, onda su jezgre $X^{\prime}_s$ i $X^{\prime}_{\tilde{s}}$ nehomeomorfne. Teorem 1.4. Ako je $1 < s < 2$ takav da $T_s$ ima beskonačnu nerekurentnu kritičnu orbitu i $f : X^{\prime}_s \to X^{\prime}_s$ je homeomorfizam, onda postoji $R \in \mathbb{Z}$ takav da su $f$ i $\sigma^R$ izotopni. U drugom dijelu teze proučavamo neekvivalentna planarna smještenja lančastih kontinuuma u punoj općenitosti. Koristimo metodu permutacija grafova veznih preslikavanja i pomoću toga pokazujemo sljedeći teorem. Teorem 1.7. Svaki lančasti kontinuum koji sadrži indekompozabilni potkontinuum se može smjestiti u ravninu na neprebrojivo mnogo jako neekvivalentnih načina. Kažemo da su smještenja $\varphi, \psi$ jako neekvivalentna ako se $\varphi \circ \psi^-1$ može proširiti do homeomorfizma ravnine. Smještenja su slabo neekvivalentna ako postoji homeomorfizam $\varphi(X) \to \psi(X)$ koji se može proširiti do homeomorfizma ravnine. Mayer je 1982. godine pitao može li se svaki indekompozabilni lančasti kontinuum smjestiti u ravninu na neprebrojivo mnogo (slabo!) neekvivalentnih načina. Dajemo pozitivan odgovor na to pitanje u slučaju unimodalnih inverznih limesa. Bavimo se i pitanjem dostupnosti točaka. Točka $x$ planarnog kontinuuma $X$ je dostupna ako postoji luk $A \subset \mathbb{R}^2$ takav da je $A \cap X = \{x\}$. Konkretno se bavimo pitanjem Nadlera i Quinna iz 1972. i pokazujemo sljedeći teorem. Teorem 1.9. Ako je $X$ lančasti kontinuum i $x \in X$ takva da niti jedna projekcija nije u cik-caku veznog preslikavanja, onda postoji smještenje od $X$ u ravninu u kojem je $x$ dostupna. Na kraju razmatramo otvorena pitanja i dajemo moguće korake prema njihovom rješenju