Diophantine Condition Research Articles

On Roth type conditions, duality and central Birkhoff sums for i.e.m.

We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the \emph{dual Roth type} condition is a condition on the \emph{backward} rotation number of a translation surface. We show that results on the cohomological equation previously proved in \cite{MY} for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted \emph{absolute} Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with \emph{subpolynomial} deviations of ergodic averages (corresponding to relative homology classes) \emph{distributional} limit shapes, which are constructed in a similar way to the \emph{limit shapes} of Birkhoff sums associated in \cite{MMY3} to functions which correspond to positive Lyapunov exponents.

On Roth type conditions, duality and central Birkhoff sums for i.e.m.

We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the \emph{dual Roth type} condition is a condition on the \emph{backward} rotation number of a translation surface. We show that results on the cohomological equation previously proved in \cite{MY} for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted \emph{absolute} Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with \emph{subpolynomial} deviations of ergodic averages (corresponding to relative homology classes) \emph{distributional} limit shapes, which are constructed in a similar way to the \emph{limit shapes} of Birkhoff sums associated in \cite{MMY3} to functions which correspond to positive Lyapunov exponents.

Rigidity of Some Abelian-by-Cyclic Solvable Group Actions on $${\mathbb {T}}^N$$

In this paper, we study a natural class of groups that act as affine transformations of $${\mathbb {T}}^N$$ . We investigate whether these solvable, “abelian-by-cyclic," groups can act smoothly and nonaffinely on $${\mathbb {T}}^N$$ while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine conditions on the action are met. In dimension two, under natural dynamical hypotheses, we get a complete classification of such actions; namely, any such group action by $$C^r$$ diffeomorphisms can be conjugated to the affine action by $$C^{r-\varepsilon }$$ conjugacy. Next, we show that in any dimension, $$C^1$$ small perturbations can be conjugated to an affine action via $$C^{1+\varepsilon }$$ conjugacy. The method is a generalization of the Herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics.

Time-changes of Heisenberg nilflows

We consider the three dimensional Heisenberg nilflows. Under a full measure set Diophantine condition on the generator of the flow we construct Bufetov functionals which are asymptotic to ergodic integrals for sufficiently smooth functions, have a modular property and scale exactly under the renor- malization dynamics. We then prove analyticity of the functionals in the transverse directions to the flow. As a consequence of this analyticity property we derive that there exists a full measure set of nilflows such that generic (non-trivial) time-changes are mixing and moreover have a stretched decay of correlations for sufficiently smooth functions. Moreover we also prove that there exists a full Hausdorff dimension set of nilflows such that generic non-trivial time-changes have polynomial decay of correlations.

Time-changes of Heisenberg nilflows

We consider the three dimensional Heisenberg nilflows. Under a full measure set Diophantine condition on the generator of the flow we construct Bufetov functionals which are asymptotic to ergodic integrals for sufficiently smooth functions, have a modular property and scale exactly under the renor- malization dynamics. We then prove analyticity of the functionals in the transverse directions to the flow. As a consequence of this analyticity property we derive that there exists a full measure set of nilflows such that generic (non-trivial) time-changes are mixing and moreover have a stretched decay of correlations for sufficiently smooth functions. Moreover we also prove that there exists a full Hausdorff dimension set of nilflows such that generic non-trivial time-changes have polynomial decay of correlations.

Hausdorff dimension of a class of three-interval exchange maps

<p style='text-indent:20px;'>In [<xref ref-type="bibr" rid="b5">5</xref>] Bourgain proves that Sarnak's disjointness conjecture holds for a certain class of three-interval exchange maps. In the present paper we slightly improve the Diophantine condition of Bourgain and estimate the constants in the proof. We further show that the new parameter set has positive, but not full Hausdorff dimension. This, in particular, implies that the Lebesgue measure of this set is zero.

Existence of whiskered KAM tori of conformally symplectic systems

Many physical problems are described by conformally symplectic systems (i.e. systems whose evolution in time transforms a symplectic form into a multiple of itself). We study the existence of whiskered tori in a family of conformally symplectic maps depending on parameters (often called drifts). We recall that whiskered tori are tori on which the motion is a rotation, but they have as many expanding/contracting directions as allowed by the preservation of the geometric structure.Our main result is formulated in an a posteriori format. We fix satisfying Diophantine conditions. We assume that we are given (1) a value of the parameter , (2) an embedding of the torus K0 into the phase space, approximately invariant under in the sense that (where is the shift by ) is small (in some norm), (3) a splitting of the tangent space at the range of K0, into three bundles which are approximately invariant under and such that the derivative satisfies ‘rate conditions’ on each of the components.Then, if some non-degeneracy conditions (verifiable by a finite calculation on the approximate solution and which do not require any global property of the map) are satisfied, we show that there is another parameter , an embedding and splittings close to the original ones which are invariant under . We also bound , and the distance of the initial and final splittings in terms of the initial error.We allow that the stable/unstable bundles are nontrivial (i.e. not homeomorphic to a product bundle). On the other hand, we show that the geometric set up has the global consequence that the center bundle is necessarily trivial (i.e. homeomorphic to a product bundle).The proof of the main theorem consists in describing an iterative process that takes advantage of cancellations coming from the geometry. Then, we show that the process converges to a true solution when started from an approximate enough solution. The iterative process leads to an efficient algorithm that is quite practical to implement.The a posteriori format of the theorem implies the usual formulation of persistence under perturbations (the solutions for the original systems are approximate solutions for the perturbation), but it also allows to justify approximate solutions produced by any method (for example numerical solutions or asymptotic formal expansions). As an application, we study the singular problem of effects of small dissipation on whiskered tori. We develop formal (presumably not convergent) expansions in the perturbative parameter (which generates dissipation) and use them as input for the a posteriori theorem. This allows to obtain lower bounds for the domain of analyticity of the tori as function of the perturbative parameter.Even if we state only the theory for maps, our results apply also to flows.

An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

We study stability times for a family of parameter dependent nonlinear Schrodinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively.

Forced quasi-periodic oscillations in strongly dissipative systems of any finite dimension

We consider a class of singular ordinary differential equations describing analytic systems of arbitrary finite dimension, subject to a quasi-periodic forcing term and in the presence of dissipation. We study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term, in the case of large dissipation. We assume the system to be conservative in the absence of dissipation, so that the forcing term is — up to the sign — the gradient of a potential energy, and both the mass and damping matrices to be symmetric and positive definite. Further, we assume a non-degeneracy condition on the forcing term, essentially that the time-average of the potential energy has a strict local minimum. On the contrary, no condition is assumed on the forcing frequency; in particular, we do not require any Diophantine condition. We prove that, under the assumptions above, a response solution always exists provided the dissipation is strong enough. This extends results previously available in the literature in the one-dimensional case.

Linearization of Gevrey flows on [formula omitted] with a Brjuno type arithmetical condition

We show that in the Gevrey topology, a d-torus flow close enough to linear with a unique rotation vector ω is linearizable as long as ω satisfies a novel Brjuno type diophantine condition. The proof is based on the fast convergence under renormalization of the associated Gevrey vector field. It requires a multidimensional continued fractions expansion of ω, and the corresponding characterization of the Brjuno type vectors. This demonstrates that renormalization methods deal very naturally with Gevrey regularity expressed in the decay of Fourier coefficients. In particular, they provide new linearization results including frequencies beyond diophantine in non-analytic topologies.

Exponential and sub-exponential stability times for the NLS on the circle

In this note we study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we state a rather flexible Birkho¤ Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively. Complete proofs are given elsewhere (see [BMP18]).

Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles

An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given dimension and endow it with the uniform norm. Assume that the translation vector satisfies a generic Diophantine condition. We establish large deviation type estimates for the iterates of such cocycles, which, moreover, are stable under small perturbations of the cocycle. As a consequence of these uniform estimates, we obtain continuity properties of the Lyapunov exponents regarded as functions on this space of cocycles. This result builds upon our previous work on this topic and its proof uses an abstract continuity theorem for the Lyapunov exponents which we derived in a recent monograph. The new feature of this paper is extending the availability of such results to cocycles that are identically singular (i.e. non-invertible everywhere), in the several variables torus translation setting. This feature is exactly what allows us, through a simple limiting argument, to obtain criteria for the positivity and simplicity of the Lyapunov exponents of such cocycles. Specializing to the family of cocycles corresponding to a block Jacobi operator, we derive consequences on the continuity, positivity and simplicity of its Lyapunov exponents, and on the continuity of its integrated density of states.

Primes with restricted digits

Let a_0in {0,ldots ,9}. We show there are infinitely many prime numbers which do not have the digit a_0 in their decimal expansion. The proof is an application of the Hardy–Littlewood circle method to a binary problem, and rests on obtaining suitable ‘Type I’ and ‘Type II’ arithmetic information for use in Harman’s sieve to control the minor arcs. This is obtained by decorrelating Diophantine conditions which dictate when the Fourier transform of the primes is large from digital conditions which dictate when the Fourier transform of numbers with restricted digits is large. These estimates rely on a combination of the geometry of numbers, the large sieve and moment estimates obtained by comparison with a Markov process.

On the CLT for rotations and BV functions

Let x↦x+α be a rotation on the circle and let φ be a step function. Denote by φn(x) the ergodic sums ∑j=0n−1φ(x+jα). For α in a class containing the rotations with bounded partial quotients and under a Diophantine condition on the discontinuities of φ, we show that φn/‖φn‖2 is asymptotically Gaussian for n in a set of density 1.

Almost-prime times in horospherical flows on the space of lattices

An integer is called almost-prime if it has fewer than a fixed number of prime factors. In this paper, we study the asymptotic distribution of almost-prime entries in horospherical flows on \begin{document}$ \Gamma\backslash {{\rm{SL}}}_n(\mathbb{R}) $\end{document} , where \begin{document}$ \Gamma $\end{document} is either \begin{document}$ {{\rm{SL}}}_n(\mathbb{Z}) $\end{document} or a cocompact lattice. In the cocompact case, we obtain a result that implies density for almost-primes in horospherical flows where the number of prime factors is independent of basepoint, and in the space of lattices we show the density of almost-primes in abelian horospherical orbits of points satisfying a certain Diophantine condition. Along the way we give an effective equidistribution result for arbitrary horospherical flows on the space of lattices, as well as an effective rate for the equidistribution of arithmetic progressions in abelian horospherical flows.

Möbius disjointness for interval exchange transformations on three intervals

We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.

Homogeneity of the spectrum for quasi-periodic Schrödinger operators

We consider the one-dimensional discrete Schrödinger operator \bigl[H(x,\omega)\varphi\bigr](n)\equiv -\varphi(n-1)-\varphi(n+1) + V(x + n\omega)\varphi(n)\ , n \in \mathbb Z , x,\omega \in [0, 1] with real-analytic potential V(x) . Assume L(E,\omega)&gt;0 for all E . Let \mathcal{S}_\omega be the spectrum of H(x,\omega) . For all \omega obeying the Diophantine condition \omega \in \mathbb{T}_{c,a} , we show the following: if \mathcal{S}_\omega \cap (E',E'')\neq \emptyset , then \mathcal{S}_\omega \cap (E',E'') is homogeneous in the sense of Carleson (see [Car83]). Furthermore, we prove, that if G_i , i=1,2 are two gaps with 1 &gt; |G_1| \ge |G_2| , then |G_2|\lesssim \exp\left(-(\log \mathrm {dist} (G_1,G_2))^A\right) , A\gg 1 . Moreover, the same estimates hold for the gaps in the spectrum on a finite interval, that is, for \mathcal S_{N,\omega}:=\cup_{x\in\mathbb T}\mathrm {spec} \; H_{[-N,N]}(x,\omega) , N \ge 1 , where H_{[-N, N]}(x, \omega) is the Schrödinger operator restricted to the interval [-N,N] with Dirichlet boundary conditions. In particular, all these results hold for the almost Mathieu operator with |\lambda| \neq 1 . For the supercritical almost Mathieu operator, we combine the methods of [GolSch08] with Jitomirskaya's approach from [Jit99] to establish most of the results from [GolSch08] with \omega obeying a strong Diophantine condition.

Escaping orbits are rare in the quasi-periodic Fermi–Ulam ping-pong

We consider the quasi-periodic Fermi–Ulam ping-pong model with no diophantine condition on the frequencies and show that typically the set of initial data which leads to escaping orbits has Lebesgue measure zero.

Multiscale KAM theorem for Hamiltonian systems

In this paper, we study the persistence of invariant tori in nearly integrable multiscale Hamiltonian systems with highorder degeneracy in the integrable part. Such Hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems. We introduce multiscale nondegenerate condition and multiscale Diophantine condition, comparable to the usual Diophantine condition. Using quasilinear KAM method, we prove a multiscale KAM theorem.

Existence of global solutions for a class of vector fields on the three-dimensional torus

This work deals with global solvability of a class of vector fields of the form L=∂/∂t+(a(x)+ib(x))(∂/∂x+λ∂/∂y), where a,b∈C∞(T1,R) and λ∈R, defined on the three-dimensional torus T3(x,y,t)≃R3/2πZ3. In addition to the interplay between the order of vanishing of the functions a and b, the change of sign of b between two consecutive zeros of a+ib has influence in the global solvability. Also, a Diophantine condition appears in a natural way in our results.