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In this paper, we give an example of a statement concerning two-player zero-sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist "paradoxical" two-player zero-sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory. [ABSTRACT FROM AUTHOR]

In this paper, we reduce pointwise convergence of polynomial ergodic averages of general measure-preserving systems acted by Z d to the case of measure-preserving systems acted by Z d with zero entropy. As an application, we can build pointwise convergence of polynomial ergodic averages for K -systems acted by Z d. [ABSTRACT FROM AUTHOR]

The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor. [ABSTRACT FROM AUTHOR]

Periodic measures are the time-periodic counterpart to invariant measures for dynamical systems and can be used to characterise the long-term periodic behaviour of stochastic systems. This paper gives sufficient conditions for the existence, uniqueness and geometric convergence of a periodic measure for time-periodic Markovian processes on a locally compact metric space in great generality. In particular, we apply these results in the context of time-periodic weakly dissipative stochastic differential equations, gradient stochastic differential equations as well as Langevin equations. We will establish the Fokker-Planck equation that the density of the periodic measure sufficiently and necessarily satisfies. Applications to physical problems shall be discussed with specific examples. [ABSTRACT FROM AUTHOR]

We examine multidimensional polynomial progressions involving linearly independent polynomials over finite fields, proving power saving bounds for sets lacking such configurations. This jointly generalises earlier results of Peluse (for the single dimensional case) and the author (for distinct degree polynomials). In contrast to the cases studied in the aforementioned two papers, a usual PET induction argument does not give Gowers norm control over multidimensional progressions that involve polynomials of the same degrees. The main challenge is therefore to obtain Gowers norm control, and we accomplish this for all multidimensional polynomial progressions with pairwise independent polynomials. The key inputs are: (1) a quantitative version of a PET induction scheme developed in ergodic theory by Donoso, Koutsogiannis, Ferré-Moragues and Sun, (2) a quantitative concatenation result for Gowers box norms in arbitrary finite abelian groups, motivated by (but different from) earlier results of Tao, Ziegler, Peluse and Prendiville; (3) an adaptation to combinatorics of the box norm smoothing technique, recently developed in the ergodic setting by the author and Frantzikinakis; and (4) a new version of the multidimensional degree lowering argument. [ABSTRACT FROM AUTHOR]

The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet, Neumann and Robin problems of harmonic analysis for the unit ball in R 3 with the corresponding behaviour of the associated ergodic inverse problems for the entire space. Rates of approximation play a basic role. The solutions themselves are evaluated by means of Fourier expansions with respect to spherical harmonics. In case of the first two problems, the basis for the investigation of the approximation and ergodic behaviour is the theory of semigroups of linear operators mapping a Banach space X into itself. The connection between the semigroup property and the major premise of Huygens' principle is emphasized. Another tool is a Drazin-like inverse operator A ad for the infinitesimal generator A of a semigroup that arises naturally in ergodic theory. This operator is a closed, not necessarily bounded, operator. It was introduced in a paper with U. Westphal (Butzer and Westphal, 1970/71) and extended to a generalized setting with J.J. Koliha (Butzer and Koliha, 2009). Unlike the latter two problems, the solution of Robin's problem does not have the semigroup property and therefore the semigroup methods applied to Dirichlet's and Neumann's problem do not work. The authors give several hints how to overcome these difficulties. [ABSTRACT FROM AUTHOR]

The simulation of chemical kinetics involving multiple scales constitutes a modeling challenge (from ordinary differential equations to Markov chains) and a computational challenge (multiple scales, large dynamical systems, time step restrictions). In this paper, we propose a new discrete stochastic simulation algorithm: the postprocessed second kind stabilized orthogonal τ -leap Runge–Kutta method (PSK- τ -ROCK). In the context of chemical kinetics, this method can be seen as a stabilization of Gillespie's explicit τ -leap algorithm combined with a postprocessor. The stabilized procedure allows to simulate problems with multiple scales (stiff), while the postprocessing procedure allows to approximate the invariant measure (e.g. mean and variance) of ergodic stochastic dynamical systems. We prove stability and accuracy of the PSK- τ -ROCK for a reference system. Numerical experiments illustrate the high reliability and efficiency of the scheme when compared to other τ -leap methods. • Optimal explicit stabilized scheme for chemical kinetics. • Inexpensive ergodic accurate postprocessor. • Explicit scheme for stiff chemical kinetics problems. • Stabilized tau-leap method. [ABSTRACT FROM AUTHOR]

Let X = G / Γ , where G is a Lie group and Γ is a lattice in G , let U be an open subset of X , and let { g t } be a one-parameter subgroup of G. Consider the set of points in X whose g t -orbit misses U ; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when G is a simple Lie group of real rank 1. In this paper we prove this conjecture for the case G = SL m + n (R) , Γ = SL m + n (Z) and g t = diag (e n t , ... , e n t , e − m t , ... , e − m t) , in fact providing an effective estimate for the codimension. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on SL m + n (R) / SL m + n (Z). We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation. [ABSTRACT FROM AUTHOR]

In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of the set of recurrent points that escape on average. Of independent interest, we prove a version of Katok's entropy formula in this non-compact setting. [ABSTRACT FROM AUTHOR]

Every affine isometric action α of a group G on a real Hilbert space gives rise to a nonsingular action α ˆ of G on the associated Gaussian probability space. In the recent paper [2] , several results on the ergodicity and Krieger type of these actions were established when the underlying orthogonal representation π of G is mixing. We develop new methods to prove ergodicity when π is only weakly mixing. We determine the type of α ˆ in full generality. Using Cantor measures, we give examples of type III 1 ergodic Gaussian actions of Z whose underlying representation is non mixing, and even has a Dirichlet measure as spectral type. We also provide very general ergodicity results for Gaussian skew product actions. [ABSTRACT FROM AUTHOR]