Michael Björklund

Biharmonic functions on groups and limit theorems for quasimorphisms along random walks

MICHAEL BJORKLUND AND TOBIAS HARTNICK¨Abstract. We show for very general classes of measures on locally compact secondcountable groups that every Borel measurable quasimorphism is at bounded distancefrom a quasi-biharmonic one. This allows us to deduce non-degenerate central limit the-orems and laws of the iterated logarithm for such quasimorphisms along regular randomwalks on topological groups using classical martingale limit theorems of Billingsley andStout. For quasi-biharmonic quasimorphism on countable groups we also obtain integralrepresentations using martingale convergence.

Aperiodic order and spherical diffraction, I: auto‐correlation of regular model sets

We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian symmetric space can be identified with a (typically non-tempered) positive-definite distribution on

THE ASYMPTOTIC SHAPE THEOREM FOR GENERALIZED FIRST PASSAGE PERCOLATION

\mathbb R^n

EQUIDISTRIBUTION OF DILATIONS OF POLYNOMIAL CURVES IN NILMANIFOLDS

. This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.We study uniform and non-uniform model sets in arbitrary locally compact second countable (lcsc) groups, which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by Meyer and used as mathematical models of quasi-crystals. We then define a notion of auto-correlation for subsets of finite local complexitiy in arbitrary lcsc groups, which generalizes Hofs classical definition beyond the class of amenable groups, and prov ide a formula for the auto-correlation of a regular model set. Along the way we show that the punctured hull of an arbitrary regular model set admits a unique invariant probability measure, even in the case where the punctured hull is non-compact and the group is non-amenable. In fact this measure is also the unique stationary measure with respect to any admissible probability measure.

Almost sure equidistribution in expansive families

This thesis consists of an introduction, a summary and 7 papers. The papers are devoted to problems in ergodic theory, equidistribution on compact manifolds and random walks on groups. In Papers A and B, we generalize two classical ergodic theorems for actions of abelian groups. The main result is a generalization of Kingman’s subadditive ergodic theorem to ergodic actions of the group Zd. In Papers C,D and E, we consider equidistribution problems on nilmanifolds. In Paper C we study the asymptotic behavior of dilations of probability measures on nilmanifolds, supported on singular sets, and prove, under some technical assumptions, effective convergences to Haar measure. In Paper D, we give a new geometric proof of an old result by Koksma on almost sure equidistribution of expansive sequences. In paper E we give necessary and sufficient conditions on a probability measure on a homogeneous Riemannian manifold to be non–atomic. Papers F and G are concerned with the asymptotic behavior of random walks on groups. In Paper F we consider homogeneous random walks on Gromov hyperbolic groups and establish a central limit theorem for random walks satisfying some technical moment conditions. Paper G is devoted to certain Bernoulli convolutions and the regularity of their value distributions.We generalize the asymptotic shape theorem in first passage percolation on

Characteristic polynomial patterns in difference sets of matrices

\mathbb{Z}^d

Almost sure absolute continuity of Bernoulli convolutions

to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of the maximal inequality for

Bohr Sets in Triple Products of Large Sets in Amenable Groups

\mathbb{Z}^d

Ergodic Theorems for Homogeneous Dilations

-cocycles of Boivin and Derriennic in the vector-valued case. This inequality will imply a very general form of Kingmans subadditive ergodic theorem. For certain classes of generalized first passage percolation, we prove further structure theorems and provide rates of convergence for the asymptotic shape theorem. We also establish a general form of the multiplicative ergodic theorem of Karlsson and Ledrappier for cocycles with values in separable Banach spaces with the Radon--Nikodym property.

Central Limit Theorems for Gromov Hyperbolic Groups

This thesis consists of an introduction, a summary and 7 papers. The papers are devoted to problems in ergodic theory, equidistribution on compact manifolds and random walks on groups. In Papers A and B, we generalize two classical ergodic theorems for actions of abelian groups. The main result is a generalization of Kingman’s subadditive ergodic theorem to ergodic actions of the group Zd. In Papers C,D and E, we consider equidistribution problems on nilmanifolds. In Paper C we study the asymptotic behavior of dilations of probability measures on nilmanifolds, supported on singular sets, and prove, under some technical assumptions, effective convergences to Haar measure. In Paper D, we give a new geometric proof of an old result by Koksma on almost sure equidistribution of expansive sequences. In paper E we give necessary and sufficient conditions on a probability measure on a homogeneous Riemannian manifold to be non–atomic. Papers F and G are concerned with the asymptotic behavior of random walks on groups. In Paper F we consider homogeneous random walks on Gromov hyperbolic groups and establish a central limit theorem for random walks satisfying some technical moment conditions. Paper G is devoted to certain Bernoulli convolutions and the regularity of their value distributions.