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Register a new userA dynamical argument for a Ramsey property
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We show by a dynamical argument that there is a positive integer valued function $q$ defined on positive integer set $mathbb N$ such that $q([log n]+1)$ is a super-polynomial with respect to positive $n$ and [liminf_{nrightarrowinfty} rleft((2n+1)^2, q(n)right)<infty,] where $r( , )$ is the opposite-Ramsey number function.
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We show that the Gurarij space $mathbb{G}$ has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. We also compute the universal minimal flow of the automorphism group of the Poulsen simplex $mathbb{P}$ and we prove that it consists of the canonical action on $mathbb{P}$ itself. This answers a question of Conley and T{o}rnquist. We show that the pointwise stabilizer of any closed proper face of $mathbb{P}$ is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simplex) is extremely amenable. These results are obtained via several Kechris-Pestov-Todorcevic correspondences, by establishing the approximate Ramsey property for several classes of finite-dimensional Banach spaces and function systems and thei
The noncommutative Gurarij space $mathbb{mathbb{mathbb{NG}}}$, initially defined by Oikhberg, is a canonical object in the theory of operator spaces. As the Fra{i}ss{e} limit of the class of finite-dimensional nuclear operator spaces, it can be seen as the noncommutative analogue of the classical Gurarij Banach space. In this paper, we prove that the automorphism group of $mathbb{mathbb{NG}}$ is extremely amenable, i.e. any of its actions on compact spaces has a fixed point. The proof relies on the Dual Ramsey Theorem, and a version of the Kechris--Pestov--Todorcevic correspondence in the setting of operator spaces. Recent work of Davidson and Kennedy, building on previous work of Arveson, Effros, Farenick, Webster, and Winkler, among others, shows that nuclear operator systems can be seen as the noncommutative analogue of Choquet simplices. The analogue of the Poulsen simplex in this context is the matrix state space $mathbb{NP}$ of the Fra{i}ss{e} limit $A(mathbb{NP})$ of the class of finite-dimensional nuclear operator systems. We show that the canonical action of the automorphism group of $mathbb{NP}$ on the compact set $mathbb{NP}_1$ of unital linear functionals on $A(mathbb{NP})$ is minimal and it factors onto any minimal action, whence providing a description of the universal minimal flow of textrm{Aut}$left( mathbb{NP}% right) $.
We define a collection of topological Ramsey spaces consisting of equivalence relations on $omega$ with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of $omega$. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) in order to obtain a dual Ramsey theorem for such ordinals.
We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $mathbf{K}$ of characteristic zero, improving results of Ingram. For that, we show that when $mathbf{K}$ is the field of rational functions of a smooth complex projective curve, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a similarity argument.
In the present paper, we study the distribution of the return points in the fibers for a RDS (random dynamical systems) nonuniformly expanding preserving an ergodic probability, we also show the abundance of nonlacunarity of hyperbolic times that are obtained along the orbits through the fibers. We conclude that any ergodic measure with positive Lyapunov exponents satisfies the nonuniform specification property between fibers. As consequences, we prove that any expanding measure is the limit of probability measure whose measures of disintegration on the fibers are supported by a finite number of return points and we prove that the average of the measures on the fibers corresponding to a disintegration, along an orbit in the base dynamics is the limit of Dirac measures supported in return orbits on the fibers.
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