The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue-Morse chain, in its realisation as a binary sequence with values in $\{\pm 1\}$. We revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure, which is related to the known representation as a Riesz product.
We find three dense subsets $Σ_I,Σ_{II}$ and $Σ_{III}$ of the spectrum of the Thue-Morse Hamiltonian, such that each energy in $Σ_I$ is extended, each energy in $Σ_{II}$ is pseudo-localized and each energy in $Σ_{III}$ is one-sided pseudo-localized. We also obtain exact estimations on the norm of the transfer matrices and the norm of the formal solutions for these energies. Especially, for $E\in Σ_{II}\cupΣ_{III}, $ the norms of the transfer matrices behave like $$ e^{c_1γ\sqrt{n}}\le\|T_{n}(E)\|\le e^{c_2γ\sqrt{n}}. $$ The local dimensions of the spectral measure on these subsets are also stu…
We study the behavior of a polynomial sequence which is defined by iterating a polynomial pair under Thue-Morse dynamic. We show that in suitable sense, the sequence will behave like $\{2\cos 2^nx: n\ge 1\}$. Basing on this property we can show that the Hausdorff dimension of the spectrum of the Thue-Morse Hamiltonian has a common positive lower bound for all coupling.
In this paper, we will provide a mathematically rigorous computer aided estimation for the exact values and robustness for Gelfond exponent of weighted Thue-Morse sequences. This result improves previous discussions on Gelfond exponent by Gelfond, Devenport, Mauduit, Rivat, Sárközy and Fan et. al.
Call a group action on a topological space \emph{biminimal} if for any points $x,y\in X$ there exists a group element taking $x$ arbitrarily close to $y$ and whose inverse takes $y$ arbitrarily close to $x$. A symbolic encoding of the Thue-Morse dynamical system is given, in terms of $ω$-automata. It is used to prove that the Thue-Morse dynamical system is minimal but not biminimal. The $ω$-automata also establish a link between Nekrashevych's presentation of limit spaces and solenoids with a construction described by Vershik and Solomyak.
We show that the images via $z\mapsto z^m$ of the continuous part of the spectral measures of the dynamical systems generated by the 0-1 sequences of the Thue-Morse type are pairwise mutually singular for different odd numbers $m\in\N$. Sarnak's conjecture on orthogonality with the Möbius function is shown to hold for such dynamical systems. The same conjecture is shown to hold for all systems induced by regular Toeplitz sequences. A non-regular Toeplitz sequence for which Sarnak's conjecture fails is constructed.
The set of unique $β$-expansions over the alphabet $\{0,1\}$ is trivial for $β$ below the golden ratio and uncountable above the Komornik-Loreti constant. Generalisations of these thresholds for three-letter alphabets were studied by Komornik, Lai and Pedicini (2011, 2017). We use S-adic words including the Thue-Morse word (which defines the Komornik-Loreti constant) and Sturmian words (which characterise generalised golden ratios) to determine the value of a certain generalisation of the Komornik-Loreti constant to three-letter alphabets.
The classic Thue--Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.
We consider the generalized Thue-Morse sequences $(t_n^{(c)})_{n\ge 0}$ ($c \in [0,1)$ being a parameter) defined by $t_n^{(c)} = e^{2πi c s_2(n)}$, where $s_2(n)$ is the sum of digits of the binary expansion of $n$. For the polynomials $σ_{N}^{(c)} (x) := \sum_{n=0}^{N-1} t_n^{(c)} e^{2πi n x}$, we have proved in [18] that the uniform norm $\|σ_N^{(c)}\|_\infty$ behaves like $N^{γ(c)}$ and the best exponent $γ(c)$ is computed. In this paper, we study the pointwise behavior and give a complete multifractal analysis of the limit $\lim_{n\to\infty}n^{-1}\log |σ_{2^n}^{(c)}(x)|$.
Given an integer $q\ge 2$ and a real number $c\in [0,1)$, consider the generalized Thue-Morse sequence $(t_n^{(q;c)})_{n\ge 0}$ defined by $t_n^{(q;c)} = e^{2πi c S_q(n)}$, where $S_q(n)$ is the sum of digits of the $q$-expansion of $n$. We prove that the $L^\infty$-norm of the trigonometric polynomials $σ_{N}^{(q;c)} (x) := \sum_{n=0}^{N-1} t_n^{(q;c)} e^{2πi n x}$, behaves like $N^{γ(q;c)}$, where $γ(q;c)$ is equal to the dynamical maximal value of $\log_q \left|\frac{\sin qπ(x+c)}{\sin π(x+c)}\right|$ relative to the dynamics $x \mapsto qx \mod 1$ and that the maximum value is attained by a…
Let $(t_n)_{n\ge0}$ be the well konwn $\pm1$ Thue-Morse sequence $$+1,-1,-1,+1,-1,+1,+1,-1,\cdots.$$ Since the 1982-1983 work of Coquet and Dekking, it is known that $\sum_{k<n}t_ke^\frac{2kπi}{3}$ is strongly related to the famous Koch curve. As a natural generalization, for integer $m\ge1$, we use $\sum_{k<n}δ_ke^\frac{2kπi}{m}$ to define generalized Koch curve, where $(δ_n)_{n\ge0}$ is the generalized Thue-Morse sequence defined to be the unique fixed point of the morphism $$+1\mapsto+1,+δ_1,\cdots,+δ_m$$ $$-1\mapsto-1,-δ_1,\cdots,-δ_m$$ beginning with $δ_0=+1$ and $δ_1,\cdots,δ_m\in\{+1,-1…
