A complex polynomial $P(z) = c_0 + c_1 z +...+ c_n z^n$ is called unimodular if $|c_j| = 1$, $j = 0,...,n$. Littlewood asked the question (1966) on how close a unimodular polynomial come to satisfying $|P(z)| \approx \sqrt{n+1}$ if $n \ge 1$? In this paper we show that for a given $0 < a < b$ and $\eps > 0$ there exist trigonometric sums $\cP(t) = n^{-1/2} \sum_{j=0}^{n-1} \exp(2πi tω(j))$ with a real frequency function $ω(j)$ which are $\eps$-flat on segment $[a,b]$ acording to the norm in $L^1([a,b])$ (as well as in $L^2([a,b])$). We apply this method to construct a dynamical system having simple spectrum and Lebesgue spectral type in the class of rank-one flows.
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…
We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due to É. Ghys, G. Hector and D. Sullivan.
We show that for any ergodic Lebesgue measure preserving transformation $f: [0,1) \rightarrow [0,1)$ and any decreasing sequence $\{b_i\}_{i=1}^{\infty}$ of positive real numbers with divergent sum, the set $$\underset{n=1}{\overset{\infty}{\cap}} \, \underset{i=n}{\overset{\infty}{\cup}}\, f^{-i}(B (R_α^{i} x,b_i))$$ has full Lebesgue measure for almost every $x \in [0,1)$ and almost every $α\in [0,1)$. Here $B(x,r)$ is the ball of radius $r$ centered at $x \in [0,1)$ and $R_α: [0,1) \rightarrow [0,1)$ is rotation by $α\in [0,1)$. As a corollary, we provide partial answer to a question asked …
The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue Orbit Equivalence, by which we understand an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non smooth Euclidean flows are shown to be Lebesgue Orbit Equivalence if and only if they admit the same number of invariant ergodic probability measures.
An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems.
Using techniques from infinite ergodic theory, Kessebohmer and Stratmann determined the asymptotic behavior of the Lebesgue measure of sets of the form $F^{-n}[α,β]$, where $[α,β]\subseteq(0,1]$ and $F$ is the Farey map. In this paper, we provide an effective version of this result, employing mostly basic properties of the transfer operator of the Farey map and an application of Freud's effective version of Karamata's Tauberian theorem.
We show that there are an irrational rotation $Tx=x+α$ on the circle $\mathbb{T}$ and a continuous $φ\colon\mathbb{T}\to\mathbb{R}$ such that for each (continuous) uniquely ergodic flow $\mathcal{S}=(S_t)_{t\in\mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{φ,\mathcal{S}}$ acting on $(X\times Y,μ\otimesν)$ by the formula $T_{φ,\mathcal{S}}(x,y)=(Tx,S_{φ(x)}(y))$, where $μ$ stands for Lebesgue measure on $\mathbb{T}$ and $ν$ denotes the unique $\mathcal{S}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mi…
We first present a modern simple proof of the classical ergodic Birkhoff's theorem and Bourgain's homogeneous bilinear ergodic theorem. This proof used the simple fact that the shift map on integers has a simple Lebesgue spectrum. As a consequence, we establish that the homogeneous bilinear ergodic averages along polynomials and polynomials in primes converge almost everywhere, that is, for any invertible measure preserving transformation $T$, acting on a probability space $(X, \mathcal{B}, μ)$, for any $f \in L^r(X,μ)$ , $g \in L^{r'}(X,μ)$ such that $\frac{1}{r}+\frac{1}{r'}= 1$, for any non…
We examine the diffraction properties of lattice dynamical systems of algebraic origin. It is well-known that diverse dynamical properties occur within this class. These include different orders of mixing (or higher-order correlations), the presence or absence of measure rigidity (restrictions on the set of possible shift-invariant ergodic measures to being those of algebraic origin), and different entropy ranks (which may be viewed as the maximal spatial dimension in which the system resembles an i.i.d.\ process). Despite these differences, it is shown that the resulting diffraction spectra a…
Let $f\colon X\to X$, $X=[0,1)$, be an ergodic IET (interval exchange transformation) relative to the Lebesgue measure on $X$. Denote by $f_t\colon X_t\to X_t$ the IET obtained by inducing $f$ to the subinterval $X=[0,t)$, $0<t<1$. We show that \[ \{0<t<1\mid f_{t} \text{is weakly mixing}\} \] is a residual subset of $X$ of full Lebesgue measure. The result is proved by establishing a generic Diophantine sufficient condition on $t$ for $f_{t}$ to be weakly mixing.
We give a simple argument that if a quasiperiodic multi-frequency Schrödinger cocycle is reducible to a constant rotation for almost all energies with respect to the density of states measure, then the spectrum of the dual operator is purely point for Lebesgue almost all values of the ergodic parameter $θ$. The result holds in the $L^2$ setting provided, in addition, that the conjugation preserves the fibered rotation number. Corollaries include localization for (long-range) 1D analytic potentials with dual ac spectrum and Diophantine frequency as well as a new result on multidimensional local…
