Siddhartha Bhattacharya

Homoclinic points and isomorphism rigidity of algebraic ℤ d -actions on zero-dimensional compact abelian groups

Letd>1, and letα andβ be mixing ℤd-actions by automorphisms of zero-dimensional compact abelian groupsX andY, respectively. By analyzing the homoclinic groups of certain sub-actions ofα andβ we prove that, if the restriction ofα to some subgroup Γ ⊂ ℤd of infinite index is expansive and has completely positive entropy, then every measurable factor mapφ: (X, α)→(Y, β) is almost everywhere equal to an affine map. The hypotheses of this result are automatically satisfied if the actionα contains an expansive automorphismαn,n ∈ ℤd, or ifα arises from a nonzero prime ideal in the ring of Laurent polynomials ind variables with coefficients in a finite prime field. Both these corollaries generalize the main theorem in [9]. In several examples we show that this kind of isomorphism rigidity breaks down if our hypotheses are weakened.

Higher order mixing and rigidity of algebraic actions on compact Abelian groups

Let Γ be a discrete group and fori=1,2; letαi be an action of Γ on a compact abelian groupXi by continuous automorphisms ofXi. We study measurable equivariant mapsf: (X1,α1)→(X2,α2), and prove a rigidity result under certain assumption on the order of mixing of the underlying actions.

Expansiveness of algebraic actions on connected groups

We study endomorphism actions of a discrete semigroup Γ on a connected group G. We give a necessary and sufficient condition for expansiveness of such actions provided G is either a Lie group or a solenoid.

Finite entropy characterizes topological rigidity on connected groups

Let X1, X2 be mixing connected algebraic dynamical systems with the Descending Chain Condition. We show that every equivariant continuous map from X1 to X2 is affine (that is, X2 is topologically rigid) if and only if the system X2 has finite topological entropy.

Isomorphism rigidity of commuting automorphisms

r. Let d > 1, and let (X, α) and (Y, β) be two zero-entropy Z d -actions on compact abelian groups by d commuting automorphisms. We show that if all lower rank subactions of α and β have completely positive entropy, then any measurable equivariant map from X to Y is an affine map. In particular, two such actions are measurably conjugate if and only if they are algebraically conjugate.

Surjunctivity and topological rigidity of algebraic dynamical systems

Let

Recent Trends in Ergodic Theory and Dynamical Systems

X

Orbit Equivalence and Topological Conjugacy of Affine Actionson Compact Abelian Groups

be a compact metrizable group and

Recent trends in ergodic theory and dynamical systems : international conference in honor of S.G. Dani's 65th birthday, Recent trends in ergodic theory and dynamical systems, December 26-29, 2012, Vadodara, India

\Gamma

Recent trends in ergodic theory and dynamical systems : international conference in honor of S.G. Dani's 65th birthday, December 26--29, 2012, Vadodara, India

a countable group acting on