/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

Introduction to the Modern Theory of Dynamical Systems: INTRODUCTION: WHAT IS LOW-DIMENSIONAL DYNAMICS?

Geometry of sets and measures in euclidean spaces fractals and rectifiability

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: Let m,n be integers which are not powers of the same integer, and let X,Y be closed subsets of the unit interval which are invariant, respectively, under times-m mod 1 and times-n mod 1. Then, for any non-zero t: dim(X+tY)=min{1,dim(X)+dim(Y)}. A similar result holds for invariant measures, and gives a simple proof of the Rudolph-Johnson theorem. Our methods also apply to many other classes of conformal fractals and measures. As another application, we extend and unify Results of Peres, Shmerkin and Nazarov, and of Moreira, concerning projections of products self-similar measures and Gibbs measures on regular Cantor sets. We show that under natural irreducibility assumptions on the maps in the IFS, the image measure has the maximal possible dimension under any linear projection other than the coordinate projections. We also present applications to Bernoulli convolutions and to the images of fractal measures under differentiable maps.

http://annals.math.princeton.edu/wp-content/uploads/annals-v175-n3-p01-p.pdf

Local entropy averages and projections of fractal measures

In this paper we shall consider a self-affine iterated function system in $\mathbb{R}^d$, d ≥ 2, where we allow a small random translation at each application of the contractions. We compute the dimension of a typical attractor of the resulting random iterated function system, complementing a famous deterministic result of Falconer, which necessarily requires restrictions on the norms of the contraction. However, our result has the advantage that we do not need to impose any additional assumptions on the norms. This is of benefit in practical applications, where such perturbations would correspond to the effect of random noise. We also give analogous results for the dimension of ergodic measures (in terms of their Lyapunov dimension). Finally, we apply our method to a problem originating in the theory of fractal image compression.

Hausdorff Dimension for Randomly Perturbed Self Affine Attractors

We investigate under which conditions a given invariant measure $\mu $ for the dynamical system defined by the Gauss map $x \mapsto 1/x \,\,{\mathrm {mod}}\,1$ is a Rajchman measure with polynomially decaying Fourier transform 

$$\begin{aligned} |\widehat{\mu }(\xi )| = O(|\xi |^{-\eta }), \quad \text {as} \quad |\xi | \rightarrow \infty . \end{aligned}$$

We show that this property holds for any Gibbs measure $\mu $ of Hausdorff dimension greater than 1 / 2 with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than 1 / 2 on badly approximable numbers, which extends the constructions of Kaufman and Queffelec–Ramare. Our main result implies that the Fourier–Stieltjes coefficients of the Minkowski’s question mark function decay to 0 polynomially answering a question of Salem from 1943. As an application of the Davenport–Erdős–LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman–Shmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents.

/pdf/fourier-transforms-of-gibbs-measures-for-the-gauss-map-4gwmimnppm.pdf

Fourier transforms of Gibbs measures for the Gauss map

We prove a multifractal formalismfor Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville-Pomeau map.

Multifractal analysis of non-uniformly hyperbolic systems

In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain assumptions the Birkhoff spectrum is real analytic. We also show that new phenomena occur; indeed, the spectrum can be constant or it can have points where it is not analytic. Conditions for these to happen are obtained. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite.

http://www.mat.uc.cl/~giommi/birkhoff22web.pdf

Multifractal analysis of Birkhoff averages for countable Markov maps

In this paper we compute the multifractal analysis for local dimensions of Bernoulli measures supported on the self-affine carpets introduced by Bedford–McMullen. This extends the work of King where the multifractal analysis is computed with strong additional separation assumptions.

/pdf/multifractal-analysis-for-bedford-mcmullen-carpets-581vv56nov.pdf

Thomas Jordan

Papers

Hausdorff Dimension for Randomly Perturbed Self Affine Attractors

Fourier transforms of Gibbs measures for the Gauss map

Multifractal analysis of non-uniformly hyperbolic systems

Multifractal analysis of Birkhoff averages for countable Markov maps

Multifractal analysis for Bedford-McMullen carpets