Related papers: Fourier decay in nonlinear dynamics

We develop a unified approach for establishing rates of decay for the Fourier transform of a wide class of dynamically defined measures. Among the key features of the method is the systematic use of the $L^2$-flattening theorem obtained in…

We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$|\widehat{\mu}(\xi)| =…

This paper relates to the Fourier decay properties of images of self-similar measures $\mu$ on $\mathbb{R}^k$ under nonlinear smooth maps $f \colon \mathbb{R}^k \to \mathbb{R}$. For example, we prove that if the linear parts of the…

We study the Fourier transforms $\widehat{\mu}(\xi)$ of non-atomic Gibbs measures $\mu$ for uniformly expanding maps $T$ of bounded distortions on $[0,1]$ or Cantor sets with strong separation. When $T$ is totally non-linear, then…

We prove that the pushforwards of a very general class of fractal measures $\mu$ on $\mathbb{R}^d$ under a large family of non-linear maps $F \colon \mathbb{R}^d \to \mathbb{R}$ exhibit polynomial Fourier decay: there exist $C,\eta>0$ such…

We show that the Fourier transform of Patterson-Sullivan measures associated to convex cocompact groups of isometries of real hyperbolic space decays polynomially quickly at infinity. The proof is based on the $L^2$-flattening theorem…

We prove the invariance of the Gibbs measure for the defocusing quintic nonlinear Schr\"odinger equation on the real line. This builds on earlier work by Bourgain, who treated the cubic nonlinearity. The key new ingredient is a growth…

We discuss a two-parameter family of maps that generalize piecewise linear, expanding maps of the circle. One parameter measures the effect of a non-linearity which bends the branches of the linear map. The second parameter rotates points…

This is a significantly expanded version of the survey paper "Mixing and decay of correlations in non-uniformly expanding maps: a survey of recent results" math/0301319. We discuss recent results on decay of correlations for non-uniformly…

We show that every self conformal measure with respect to a $C^2 (\mathbb{R})$ IFS $\Phi$ has polynomial Fourier decay under some mild and natural non-linearity conditions. In particular, every such measure has polynomial decay if $\Phi$ is…

The main result of this paper is, that if we suppose that a function is absolutely continuous and uniformly H\"older continuous and that its finite difference function does not oscillate infinitely often on a bounded interval, then the…

The increasing rate of the Birkhoff sums in the infinite iterated function systems with polynomial decay of the derivative (for example the Gauss map) is studied. For different unbounded potential functions, the Hausdorff dimensions of the…

We establish power Fourier decay for equilibrium states of parabolic $C^{1+\alpha}$ iterated function systems with overlaps satisfying a multiscale nonlinearity condition. This class includes the Lyons conductance measures $\nu_t$, $0<t<1$,…

We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from…

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has…

We establish an exponential error term for the renewal theorem in the context of products of random matrices, which is surprising compared with classical abelian cases. A key tool is the Fourier decay of the Furstenberg measures on the…

We consider a model for chaotic diffusion with amplification on graphs associated with piecewise-linear maps of the interval [S. Lepri, Chaos Solitons & Fractals, 139,110003 (2020)]. We determine the conditions for having fat-tailed…

We prove that any C^{1+} transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion…

In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these…

We consider measures supported on sets of irrational numbers possessing many consecutive partial quotients satisfying a condition based on the previous partial quotients. We show that under mild assumptions, such sets will always support…