Szemerédi’s Theorem via Ergodic Theory

Individual ergodic theorem for intuitionistic fuzzy observables using intuitionistic fuzzy state

The classical ergodic theory hasbeen built on σ-algebras. Later the Individual ergodictheorem was studied on more general structures like MV-algebrasand quantum structures. The aim of this paper is to formulate theIndividual ergodic theorem for intuitionistic fuzzy observablesusing  m-almost everywhere convergence, where  m...

Idempotent Ultrafilters, Multiple Weak Mixing and Szemerédi’s Theorem for Generalized Polynomials

It is possible to formulate the polynomial Szemerédi theorem as follows: Let qi(x) ∈ Q[x] with qi(Z) ⊂ Z, 1 ≤ i ≤ k. If E ⊂ N has positive upper density then there are a, n ∈ N such that {a, a+q1(n)−q1(0), a+qk(n)−qk(0)} ⊂ E. Using methods of abstract ergodic theory, topological algebra in βN, and some recently-obtained knowledge concerning the relationship between translations on nilmanifolds ...

A Mean Ergodic Theorem For Asymptotically Quasi-Nonexpansive Affine Mappings in Banach Spaces Satisfying Opial's Condition

Non-linear ergodic theorems in complete non-positive curvature metric spaces

Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...

A Rapidly-converging Lower Bound for the Joint Spectral Radius via Multiplicative Ergodic Theory

We use ergodic theory to prove a quantitative version of a theorem of M. A. Berger and Y. Wang, which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices. The proof rests on a theorem asserting the existence of a continuous invariant splitting for certain matrix cocycles defined over a minimal homeomorphism and having the property th...