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In this paper, we characterize topological two-fold self-joinings of faithful Cartan **Z**^{k}×**Z**_{≥ 0}^{l}-actions by endomorphisms on solenoids, when the rank *k+l* is at least 3. For example, it is proved as a special case that any topologically transitive invariant closed subset in **T**^{2} under the semigroup action generated by × 2, × 3 and × 5 is either finite, or a finite union of parallel one-dimensional affine subtori, or **T**^{2}.

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In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's Möbius disjointness conjecture holds for any system for which every invariant Borel probability measure has sub-polynomial measure complexity.
Moreover, it is proved that the following classes of topological dynamical systems (X,T) meet this condition and hence satisfy Sarnak's conjecture: (1) Each invariant Borel probability measure of T has discrete spectrum. (2) T is a homotopically trivial C^{∞} skew product system on **T**^{2} over an irrational rotation of the circle. Combining this with the previous results it implies that the Möbius disjointness conjecture holds for any C^{∞} skew product system on **T**^{2}. (3) T is a continuous skew product map of the form (ag,y+h(g)) on G×**T**^{1} over a minimal rotation of the compact metric abelian group G and T preserves a measurable section. (4) T is a tame system.

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We prove that the Möbius function is disjoint to all Lipschitz continuous skew product dynamical systems on the 3-dimensional Heisenberg nilmanifold over a minimal rotation of the 2-dimensional torus.

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We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers r(G) and m(G) associated with the roots system of the Lie algebra of a Lie group G. If the dimension of the manifold is smaller than r(G), then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most m(G), we show there is a quasi-invariant measure on the manifold such that the action is measurable isomorphic to a relatively measure preserving action over a standard boundary action.

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For a smooth action of **Z**^{d} preserving a Borel probability measure, we show that entropy satisfies a certain “product structure” along coarse unstable manifolds. Moreover, given two smooth **Z**^{d}-actions—one of which is a measurable factor of the other—we show that all coarse Lyapunov exponents contributing to the entropy of the factor system are coarse Lyapunov exponents of the total system and derive an Abramov–Rohlin formula for entropy subordinated to coarse unstable manifolds.

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In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices.
Suppose Γ is a lattice in semisimple Lie group, all of whose factors have rank 2 or higher. Let α be a smooth Γ -action on a compact nilmanifold M that lifts to an action on the universal cover. If the linear data ρ of α contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of α and ρ, on a finite-index subgroup of Γ. If α is a C^{∞} action and contains an Anosov element, then the semiconjugacy is a C^{∞} conjugacy.
As a corollary, we obtain C^{∞} global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank 2 or higher. We also obtain global rigidity of Anosov actions of SL(n,**Z**) on **T**^{n} for n≥5 and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.

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We provide a criterion for a point satisfying the required disjointness condition in Sarnak's Möbius Disjointness Conjecture. As a direct application, we have that the conjecture holds for any topological model of an ergodic system with discrete spectrum.

For τ>2, let T be a C^{τ} skew product map of the form (x+α,y+h(x)) on **T**^{2} over a rotation of the circle. We show that if T preserves a measurable section, then it is disjoint to the Möbius sequence. This in particular implies that any non-uniquely ergodic C^{τ} skew product map on **T**^{2} has a finite index factor that is disjoint to the Möbius sequence.

We show that the Möbius function is disjoint to every analytic skew product dynamical system on **T**^{2} over a rotation of the circle.

Given a **Z**^{r}-action α on a nilmanifold X by automorphisms and an ergodic α -invariant probability measure μ , we show that μ is the uniform measure on X, unless modulo finite index modification, one of the following obstructions occurs for an algebraic factor action:

1. The factor measure has zero entropy;

2. The factor action is virtually cyclic.

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We show that all C^{∞} Anosov **Z**^{r}-actions on tori and
nilmanifolds without rank-one factor actions are, up to C^{∞} conjugacy, actions by automorphisms.

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The Euclidean minimum M(K) of a number field K is an important numerical invariant that indicates whether K is
norm-Euclidean. It is the supremum of both the Euclidean spectrum Spec(K) and inhomogeneous spectrum Spec(K) of K. When
K is a non-CM field of unit rank 2 or higher, Cerri showed M(K) is isolated and attained in Spec(K) and can be computed in finite
time. We extend Cerri's works by applying recent dynamical results of Lindenstrauss and Wang. In particular, the following facts
are proved:

(1) If K is a non-CM field of unit rank 2 or higher, then the computational complexity of M(K) is bounded in terms of the degree,
discriminant and regulator of K;

(2) For any number field K of unit rank 3 or higher, M(K) is isolated and attained in Spec(K) and Cerri's algorithm computes M(K) in finite time.

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We show that if r≥3 and α is a faithful **Z**^{r}-Cartan action on a torus **T**^{d} by automorphisms, then any closed subset of **T**^{2d} which is invariant and topologically transitive under the diagonal **Z**^{r}-action α×α is homogeneous, in the sense that it is either the full torus **T**^{2d}, or a finite set of rational points, or a finite disjoint union of parallel translates of some d-dimensional invariant subtorus. A counterexample is constructed for the rank 2 case.

Berend gave necessary and sufficient conditions on a **Z**^{r} -action on a torus **T**^{d} by toral automorphisms in order for every orbit be either finite or dense. One of these conditions is that on every common eigendirection of the **Z**^{r}-action is expanded by at least one element under the action. In this paper, we investigate what happens when this condition is removed; more generally, we consider the orbit of a point under a set of elements which act in a quasi-isometric way on a given set of eigendirections. This analysis is used in an essential way in the work of the author with E. Lindenstrauss classifying topological self-joinings of maximal **Z**^{r}-actions on tori when r≥ 3.

We generalize Bourgain-Lindenstrauss-Michel-Venkatesh's recent one-dimensional quantitative density result to abelian algebraic actions on higher dimensional tori. Up to finite index, the group actions that we study are conjugate to the action of U_{K} , the group of units of some non-CM number field K, on a compact quotient of K⊗** _{Q}R** . In such a setting, we investigate how fast the orbit of a generic point can become dense in the torus. This effectivizes a special case of a theorem of Berend; and is deduced from a parallel measure-theoretical statement which effectivizes a special case of a result by Katok-Spatzier. In addition, we specify two numerical invariants of the group action that determine the quantitative behavior, which have number-theoretical significance.

Rauzy diagrams are oriented graphs that represent Rauzy-Veech inductions on interval exchange maps. To each such diagram is associated a translation surface. It was conjectured that the natural group morphism from the fundamental group of a Rauzy diagram to the mapping class group of the corresponding translation surface is surjective. We prove this is true for the special case of genus 1, i.e. when the translation surface is a flat torus with finitely many marked points.

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