This paper is a complement to [21]. It surveys the works on the Furstenberg set S = {2 m 3 n : n ... more This paper is a complement to [21]. It surveys the works on the Furstenberg set S = {2 m 3 n : n ≥ 0, m ≥ 0} and its random version T. We also present some new results. For example, it is proved that T almost surely contains a subset of positive lower density which is 4 3-Rider. It is also proved that a class of random sets of integers are Sidon sets when Bourgain's condition is not satisfied; this generalizes a result of Kahane-Katznelson. Some open questions about S and T are listed at the end of the paper.
We consider the generalized Thue-Morse sequences (t (c) n) n≥0 (c ∈ [0, 1) being a parameter) def... more We consider the generalized Thue-Morse sequences (t (c) n) n≥0 (c ∈ [0, 1) being a parameter) defined by t (c) n = e 2πics2(n) , where s 2 (n) is the sum of digits of the binary expansion of n. For the polynomials σ (c) N (x) := N −1 n=0 t (c) n e 2πinx , we have proved in [18] that the uniform norm σ (c) N ∞ 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→∞ n −1 log |σ (c) 2 n (x)|. Contents 1. Introduction and main results 1 2. Minimization for f c and exponent χ − (f c , T (b(c))) 6 3. Existence of the exponent χ(f c , T (b(c))) 12 4. Hausdorff dimension of {c : χ − (f c , T (b(c))) = −∞} 15 5. Pointwise behavior of n−1 k=0 | cos π2 k x| 22 6. Multifractal analysis of n−1 k=0 | cos π(2 k x + c)| 23 References 35
We introduce the notion of Bohr chaoticity [2], which is a topological invariant, and is opposite... more We introduce the notion of Bohr chaoticity [2], which is a topological invariant, and is opposite to the property required by Sarnak's conjecture. Such a system is by definition never orthogonal to any non-trivial weight and it must be of positive entropy. But having positive entropy is not sufficient to ensure the Bohr chaoticity [1]. Using Riesz products we prove the Bohr chaoticity for the following systems when they have positive entropy: endomorphisms on tori, subshifts of finite type, β-shifts, principal algebraic Z d-actions on the torus [2, 3]. These are joint works with Shilei FAN, Wexiao SHEN, Klaus SCHMIDT, Evgeny VERBITSKIY.
In this paper we study the Dvoretzky covering problem with nonuniformly distributed centers. When... more In this paper we study the Dvoretzky covering problem with nonuniformly distributed centers. When the probability law of the centers admits an absolutely continuous density which satisfies a regular condition on the set of essential infimum points, we give a necessary and sufficient condition for covering the circle. When the lengths of covering intervals are of the form ℓ n = c n , we give a necessary and sufficient condition for covering the circle, without imposing any regularity on the density function.
Given an integer q ≥ 2 and a real number c ∈ [0, 1), consider the generalized Thue-Morse sequence... more Given an integer q ≥ 2 and a real number c ∈ [0, 1), consider the generalized Thue-Morse sequence (t (q;c) n) n≥0 defined by t (q;c) n = e 2πicSq(n) , where S q (n) is the sum of digits of the q-expansion of n. We prove that the L ∞-norm of the trigonometric polynomials σ (q;c) N (x) := N −1 n=0 t (q;c) n e 2πinx , behaves like N γ(q;c) , where γ(q; c) is equal to the dynamical maximal value of log q sin qπ(x+c) sin π(x+c) relative to the dynamics x → qx mod 1 and that the maximum value is attained by a q-Sturmian measure. Numerical values of γ(q; c) can be computed. Contents 1. Introduction and main results 1 2. General setting of maximization and minimization 6 3. Gelfond exponent and maximization problem 9 4. Maximization for f c and Sturmian measures 11 5. Pre-Sturmian condition implies Sturmian condition 17 6. Appendix A: q-Sturmian measures 27 7. Appendix B: Computation of β(c) and γ(c) 30 References 36
We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exp... more We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than$\frac{1}{2}$are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of strong Gelfond property are well controlled almost everywhere. We prove that for any$q$-multiplicative sequence, the Gelfond property implies the strong Gelfond property and that sequences realized by dynamical systems can be fully oscillating and have the Gelfond property.
We study properties of stationary determinantal point processes X on Z from different points of v... more We study properties of stationary determinantal point processes X on Z from different points of views. It is proved that X ∩ N is almost surely Bohr-dense and good universal for almost everywhere convergence in L 1 , and that X is not syndetic but X + X = Z. For the associated centered random field, we obtain a sub-Gaussian property, a Salem-Littlewood inequality and a Khintchine-Kahane inequality. Results can be generalized to Z d .
Abstract: We study Ruelle operators on expanding and mixing dynamical systems with potential func... more Abstract: We study Ruelle operators on expanding and mixing dynamical systems with potential function satisfying the Dini condition. We give an estimate for the convergence speed of the iterates of a Ruelle operator. Our proof avoids Markov partitions. This is the second part of our research on Ruelle operators. 1.
In this article, we prove that a compact open set in the field $\mathbb{Q}_p$ of $p$-adic numbers... more In this article, we prove that a compact open set in the field $\mathbb{Q}_p$ of $p$-adic numbers is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation, and also if and only if it is $p$-homogeneous which is easy to check. We also characterize spectral sets in $\mathbb{Z}/p^n \mathbb{Z}$ ($p\ge 2$ prime, $n\ge 1$ integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in $\mathbb{Q}_p$, some of which are self-similar measures.
In this article, we prove that a compact open set in the field Q_p of p-adic numbers is a spectra... more In this article, we prove that a compact open set in the field Q_p of p-adic numbers is a spectral set if and only if it tiles Q_p by translation, and also if and only if it is p-homogeneous which is easy to check. We also characterize spectral sets in Z/p^n Z (p> 2 prime, n> 1 integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in Q_p, some of which are self-similar measures.
We define oscillating sequences which include the Möbius function in the number theory. We also d... more We define oscillating sequences which include the Möbius function in the number theory. We also define minimally mean attractable flows and minimally mean-L-stable flows. It is proved that all oscillating sequences are linearly disjoint from minimally mean attractable and minimally mean-L-stable flows. In particular, that is the case for the Möbius function. Several minimally mean attractable and minimally mean-L-stable flows are examined. These flows include the ones defined by all p-adic polynomials, all p-adic rational maps with good reduction, all automorphisms of 2-torus with zero topological entropy, all diagonalized affine maps of 2-torus with zero topological entropy, all Feigenbaum zero topological entropy flows, and all orientation-preserving circle homeomorphisms.
We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set o... more We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers $S=\{2^{m}3^{n}\}$ and compare them to those of its random analogue $T$. In this half-expository work, we show for example that $S$ is "Khinchin distributed", is far from being Hartman-distributed while $T$ is, and that $S$ is a $\Lambda(p)$ set for all $2<p<\infty$ and that $T$ is a $p$-Rider set for all $p$ such that $4/3<p<2$. Measure-theoretic and probabilistic techniques, notably martingales, play an important role in this work.
The random trigonometric series ∑∞ n=1 ρn cos(nt + ωn) on the circle T are studied under the cond... more The random trigonometric series ∑∞ n=1 ρn cos(nt + ωn) on the circle T are studied under the conditions ∑ |ρn| =∞ and ρn → 0, where {ωn} are iid and uniformly distributed on T. They are almost surely not Fourier-Stieljes series but define pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which produces a class of random measures. The behaviors of the partial sums of the above series are proved to be multifractal. Our theory holds on the torus T of dimension d ≥ 1.
We give a survey on some recent developments in the spectral theory of transfer operators, also c... more We give a survey on some recent developments in the spectral theory of transfer operators, also called Ruelle-Perron-Frobenius (RPF) operators, associated to expanding and mixing dynamical systems. Different methods for spectral study are presented. Topics include maximal eigenvalue of RPF operators, smooth invariant measures, ergodic theory for chain of markovian projections, equilibrium states, spectral gaps for RPF operators, spectral decomposition and perturbation theory, central limit theorem, Hilbert metric and convergence speeds of RPF operators, and dynamical determinants and zeta functions.
For a continuous N or Z action on a compact space, we introduce the notion of Bohr chaoticity, wh... more For a continuous N or Z action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic Z actions of positive entropy are Bohr-chaotic. The same is proved for principal algebraic Z (d ≥ 2) actions of positive entropy under the condition of existence of summable homoclinic points.
For a totally uniquely ergodic dynamical system, we prove a topological Wiener-Wintner ergodic th... more For a totally uniquely ergodic dynamical system, we prove a topological Wiener-Wintner ergodic theorem with polynomial weights under the coincidence of the quasi discrete spectrums of the system in both senses of Abramov and of Hahn-Parry. The result applies to ergodic nilsystems. Fully oscillating sequences can then be constructed on nilmanifolds.
A homographic map in the field of p-adic numbers Qp is studied as a dynamical system on P 1 (Qp),... more A homographic map in the field of p-adic numbers Qp is studied as a dynamical system on P 1 (Qp), the projective line over Qp. If such a system admits one or two fixed points in Qp, then it is conjugate to an affine dynamics whose dynamical structure has been investigated by Fan and Fares [16]. In this paper, we shall mainly solve the remaining case that the system admits no fixed point. We shall prove that this system can be decomposed into a finite number of minimal subsystems which are topologically conjugate to each other. All the minimal subsystems are exhibited and the unique invariant measure for each minimal subsystem is determined.
A polynomial of degree ≥ 2 with coefficients in the ring of padic numbers Zp is studied as a dyna... more A polynomial of degree ≥ 2 with coefficients in the ring of padic numbers Zp is studied as a dynamical system on Zp. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on Z 2 , we exhibit all its minimal subsystems.
Let p 2 be a prime number and let Z p be the ring of all p-adic integers. For all α, β, z ∈ Z p ,... more Let p 2 be a prime number and let Z p be the ring of all p-adic integers. For all α, β, z ∈ Z p , define T α,β (z) = αz + β. It is shown that the dynamical system (Z p , T α,β) is minimal if and only if α ∈ 1 + p r p Z p and β is a unit, here r p = 1 (respectively r p = 2) if p 3 (respectively if p = 2), and that when it is minimal, it is strictly ergodic and topologically conjugate to (Z p , T 1,1) with an analytic and isometric conjugacy. More importantly, when the system is not minimal, we find all its strictly ergodic components. As application, monomial systems S n,ρ (z) = ρz n on the group 1 + pZ p are also discussed.
This paper is a complement to [21]. It surveys the works on the Furstenberg set S = {2 m 3 n : n ... more This paper is a complement to [21]. It surveys the works on the Furstenberg set S = {2 m 3 n : n ≥ 0, m ≥ 0} and its random version T. We also present some new results. For example, it is proved that T almost surely contains a subset of positive lower density which is 4 3-Rider. It is also proved that a class of random sets of integers are Sidon sets when Bourgain's condition is not satisfied; this generalizes a result of Kahane-Katznelson. Some open questions about S and T are listed at the end of the paper.
We consider the generalized Thue-Morse sequences (t (c) n) n≥0 (c ∈ [0, 1) being a parameter) def... more We consider the generalized Thue-Morse sequences (t (c) n) n≥0 (c ∈ [0, 1) being a parameter) defined by t (c) n = e 2πics2(n) , where s 2 (n) is the sum of digits of the binary expansion of n. For the polynomials σ (c) N (x) := N −1 n=0 t (c) n e 2πinx , we have proved in [18] that the uniform norm σ (c) N ∞ 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→∞ n −1 log |σ (c) 2 n (x)|. Contents 1. Introduction and main results 1 2. Minimization for f c and exponent χ − (f c , T (b(c))) 6 3. Existence of the exponent χ(f c , T (b(c))) 12 4. Hausdorff dimension of {c : χ − (f c , T (b(c))) = −∞} 15 5. Pointwise behavior of n−1 k=0 | cos π2 k x| 22 6. Multifractal analysis of n−1 k=0 | cos π(2 k x + c)| 23 References 35
We introduce the notion of Bohr chaoticity [2], which is a topological invariant, and is opposite... more We introduce the notion of Bohr chaoticity [2], which is a topological invariant, and is opposite to the property required by Sarnak's conjecture. Such a system is by definition never orthogonal to any non-trivial weight and it must be of positive entropy. But having positive entropy is not sufficient to ensure the Bohr chaoticity [1]. Using Riesz products we prove the Bohr chaoticity for the following systems when they have positive entropy: endomorphisms on tori, subshifts of finite type, β-shifts, principal algebraic Z d-actions on the torus [2, 3]. These are joint works with Shilei FAN, Wexiao SHEN, Klaus SCHMIDT, Evgeny VERBITSKIY.
In this paper we study the Dvoretzky covering problem with nonuniformly distributed centers. When... more In this paper we study the Dvoretzky covering problem with nonuniformly distributed centers. When the probability law of the centers admits an absolutely continuous density which satisfies a regular condition on the set of essential infimum points, we give a necessary and sufficient condition for covering the circle. When the lengths of covering intervals are of the form ℓ n = c n , we give a necessary and sufficient condition for covering the circle, without imposing any regularity on the density function.
Given an integer q ≥ 2 and a real number c ∈ [0, 1), consider the generalized Thue-Morse sequence... more Given an integer q ≥ 2 and a real number c ∈ [0, 1), consider the generalized Thue-Morse sequence (t (q;c) n) n≥0 defined by t (q;c) n = e 2πicSq(n) , where S q (n) is the sum of digits of the q-expansion of n. We prove that the L ∞-norm of the trigonometric polynomials σ (q;c) N (x) := N −1 n=0 t (q;c) n e 2πinx , behaves like N γ(q;c) , where γ(q; c) is equal to the dynamical maximal value of log q sin qπ(x+c) sin π(x+c) relative to the dynamics x → qx mod 1 and that the maximum value is attained by a q-Sturmian measure. Numerical values of γ(q; c) can be computed. Contents 1. Introduction and main results 1 2. General setting of maximization and minimization 6 3. Gelfond exponent and maximization problem 9 4. Maximization for f c and Sturmian measures 11 5. Pre-Sturmian condition implies Sturmian condition 17 6. Appendix A: q-Sturmian measures 27 7. Appendix B: Computation of β(c) and γ(c) 30 References 36
We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exp... more We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than$\frac{1}{2}$are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of strong Gelfond property are well controlled almost everywhere. We prove that for any$q$-multiplicative sequence, the Gelfond property implies the strong Gelfond property and that sequences realized by dynamical systems can be fully oscillating and have the Gelfond property.
We study properties of stationary determinantal point processes X on Z from different points of v... more We study properties of stationary determinantal point processes X on Z from different points of views. It is proved that X ∩ N is almost surely Bohr-dense and good universal for almost everywhere convergence in L 1 , and that X is not syndetic but X + X = Z. For the associated centered random field, we obtain a sub-Gaussian property, a Salem-Littlewood inequality and a Khintchine-Kahane inequality. Results can be generalized to Z d .
Abstract: We study Ruelle operators on expanding and mixing dynamical systems with potential func... more Abstract: We study Ruelle operators on expanding and mixing dynamical systems with potential function satisfying the Dini condition. We give an estimate for the convergence speed of the iterates of a Ruelle operator. Our proof avoids Markov partitions. This is the second part of our research on Ruelle operators. 1.
In this article, we prove that a compact open set in the field $\mathbb{Q}_p$ of $p$-adic numbers... more In this article, we prove that a compact open set in the field $\mathbb{Q}_p$ of $p$-adic numbers is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation, and also if and only if it is $p$-homogeneous which is easy to check. We also characterize spectral sets in $\mathbb{Z}/p^n \mathbb{Z}$ ($p\ge 2$ prime, $n\ge 1$ integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in $\mathbb{Q}_p$, some of which are self-similar measures.
In this article, we prove that a compact open set in the field Q_p of p-adic numbers is a spectra... more In this article, we prove that a compact open set in the field Q_p of p-adic numbers is a spectral set if and only if it tiles Q_p by translation, and also if and only if it is p-homogeneous which is easy to check. We also characterize spectral sets in Z/p^n Z (p> 2 prime, n> 1 integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in Q_p, some of which are self-similar measures.
We define oscillating sequences which include the Möbius function in the number theory. We also d... more We define oscillating sequences which include the Möbius function in the number theory. We also define minimally mean attractable flows and minimally mean-L-stable flows. It is proved that all oscillating sequences are linearly disjoint from minimally mean attractable and minimally mean-L-stable flows. In particular, that is the case for the Möbius function. Several minimally mean attractable and minimally mean-L-stable flows are examined. These flows include the ones defined by all p-adic polynomials, all p-adic rational maps with good reduction, all automorphisms of 2-torus with zero topological entropy, all diagonalized affine maps of 2-torus with zero topological entropy, all Feigenbaum zero topological entropy flows, and all orientation-preserving circle homeomorphisms.
We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set o... more We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers $S=\{2^{m}3^{n}\}$ and compare them to those of its random analogue $T$. In this half-expository work, we show for example that $S$ is "Khinchin distributed", is far from being Hartman-distributed while $T$ is, and that $S$ is a $\Lambda(p)$ set for all $2<p<\infty$ and that $T$ is a $p$-Rider set for all $p$ such that $4/3<p<2$. Measure-theoretic and probabilistic techniques, notably martingales, play an important role in this work.
The random trigonometric series ∑∞ n=1 ρn cos(nt + ωn) on the circle T are studied under the cond... more The random trigonometric series ∑∞ n=1 ρn cos(nt + ωn) on the circle T are studied under the conditions ∑ |ρn| =∞ and ρn → 0, where {ωn} are iid and uniformly distributed on T. They are almost surely not Fourier-Stieljes series but define pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which produces a class of random measures. The behaviors of the partial sums of the above series are proved to be multifractal. Our theory holds on the torus T of dimension d ≥ 1.
We give a survey on some recent developments in the spectral theory of transfer operators, also c... more We give a survey on some recent developments in the spectral theory of transfer operators, also called Ruelle-Perron-Frobenius (RPF) operators, associated to expanding and mixing dynamical systems. Different methods for spectral study are presented. Topics include maximal eigenvalue of RPF operators, smooth invariant measures, ergodic theory for chain of markovian projections, equilibrium states, spectral gaps for RPF operators, spectral decomposition and perturbation theory, central limit theorem, Hilbert metric and convergence speeds of RPF operators, and dynamical determinants and zeta functions.
For a continuous N or Z action on a compact space, we introduce the notion of Bohr chaoticity, wh... more For a continuous N or Z action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic Z actions of positive entropy are Bohr-chaotic. The same is proved for principal algebraic Z (d ≥ 2) actions of positive entropy under the condition of existence of summable homoclinic points.
For a totally uniquely ergodic dynamical system, we prove a topological Wiener-Wintner ergodic th... more For a totally uniquely ergodic dynamical system, we prove a topological Wiener-Wintner ergodic theorem with polynomial weights under the coincidence of the quasi discrete spectrums of the system in both senses of Abramov and of Hahn-Parry. The result applies to ergodic nilsystems. Fully oscillating sequences can then be constructed on nilmanifolds.
A homographic map in the field of p-adic numbers Qp is studied as a dynamical system on P 1 (Qp),... more A homographic map in the field of p-adic numbers Qp is studied as a dynamical system on P 1 (Qp), the projective line over Qp. If such a system admits one or two fixed points in Qp, then it is conjugate to an affine dynamics whose dynamical structure has been investigated by Fan and Fares [16]. In this paper, we shall mainly solve the remaining case that the system admits no fixed point. We shall prove that this system can be decomposed into a finite number of minimal subsystems which are topologically conjugate to each other. All the minimal subsystems are exhibited and the unique invariant measure for each minimal subsystem is determined.
A polynomial of degree ≥ 2 with coefficients in the ring of padic numbers Zp is studied as a dyna... more A polynomial of degree ≥ 2 with coefficients in the ring of padic numbers Zp is studied as a dynamical system on Zp. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on Z 2 , we exhibit all its minimal subsystems.
Let p 2 be a prime number and let Z p be the ring of all p-adic integers. For all α, β, z ∈ Z p ,... more Let p 2 be a prime number and let Z p be the ring of all p-adic integers. For all α, β, z ∈ Z p , define T α,β (z) = αz + β. It is shown that the dynamical system (Z p , T α,β) is minimal if and only if α ∈ 1 + p r p Z p and β is a unit, here r p = 1 (respectively r p = 2) if p 3 (respectively if p = 2), and that when it is minimal, it is strictly ergodic and topologically conjugate to (Z p , T 1,1) with an analytic and isometric conjugacy. More importantly, when the system is not minimal, we find all its strictly ergodic components. As application, monomial systems S n,ρ (z) = ρz n on the group 1 + pZ p are also discussed.
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