The space of invariant measures for countable Markov shifts

We introduce two properties of dynamical systems on Polish metric spaces: closeability and linkability. We show that they imply density of ergodic measures in the space of invariant probability measures and the existence of a generic point for every invariant measure. In the compact case, it follows from our conditions that the set of invariant measures is either a singleton of a measure concentrated on a periodic orbit or the Poulsen simplex. We provide examples showing that closability and linkability are independent properties. Our theory applies to systems with the periodic specification property, irreducible Markov chains over a countable alphabet, certain coded systems including $\unicode[STIX]{x1D6FD}$-shifts and $S$-gap shifts, $C^{1}$-generic diffeomorphisms of a compact manifold $M$ and certain geodesic flows of a complete connected negatively curved manifold.

Markov finite approximation of Frobenius-Perron operator

Choquet simplices as spaces of invariant probability measures on post-critical sets

Abstract. In this paper we investigate an optimal control problem in the space of measures on R2. The problem is motivated by a stochastic interacting particle model which gives the 2-D Navier-Stokes equations in their vorticity formulation as a mean-field equation. We prove that the associated HamiltonJacobi-Bellman equation, in the space of probability measures, is well posed in an appropriately defined viscosity solution sense.

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpinski gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.

The goal of the present chapter is to present in a self-contained manner, elements of differential calculus and stochastic analysis over spaces of probability measures. Such a calculus will play a crucial role in the sequel when we discuss stochastic control of dynamics of the McKean-Vlasov type, and various forms of the master equation for mean field games. After reviewing the standard metric theory of spaces of probability measures, we introduce a notion of differentiability of functions of measures tailor-made to our needs. We provide a thorough analysis of its properties, and relate it to different notions of differentiability which have been used in the existing literature, in particular the geometric notion of Wasserstein gradient. Finally, we derive a first form of chain rule (Ito’s formula) for functions of flows of measures, and we illustrate its versatility on a couple of applications.

We construct logistic maps whose restriction to the ω-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor system corresponding to the restriction of a logistic map to the ω-limit set of its critical point. Furthermore, we show that such a logistic map f can be taken so that each such invariant measure has zero Lyapunov exponent and is an equilibrium state of f for the potential −ln |f′|.

In this paper we study the stability of the replicator dynamics for symmetric games when the strategy set is a separable metric space. In this case the replicator dynamics evolves in a space of measures. We study stability criteria with respect to different topologies and metrics on the space of probability measures. This allows us to establish relations among Nash equilibria (of a certain normal form game) and the stability of the replicator dynamics in different metrics. Some examples illustrate our results.

The actions induced by a random dynamical system on spaces of probability measures on the state space are investigated, and generalisations of the notion of an attractor are discussed and compared. For the particular case of a random dynamical system generated by a stochastic differential equation the notion of an attractor for the associated Markov semigroup had previously been discussed in several instances in the literature. It is re-discovered here as a special case of a more general notion of an attractor in the space of measures.

Mayer control problem with probabilistic uncertainty on initial positions

Ensemble Control of Dynamic Systems with Square Impulses

Let S be a non-exceptional oriented surface of finite type. We discuss the action of subgroups of the mapping class group of S on the CAT(0)-boundary of the completion of Teichmueller space with respect to the Weil-Petersson metric. We show that the set of invariant Borel probability measures for the Weil-Petersson flow on moduli space which are supported on closed orbits is dense in the space of invariant Borel probability measures.

In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [Y. Pesin and S. Senti. Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J.5(3) (2005), 669–678; Y. Pesin and S. Senti. Equilibrium measures for maps with inducing schemes. Preprint, 2007]. We show that under some natural assumptions on the inducing schemes—which hold for many known examples—any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [J. Buzzi. Markov extensions for multi-dimensional dynamical systems. Israel J. Math.112 (1999), 357–380], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [H. Bruin. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics. Comm. Math. Phys.168(3) (1995), 571–580] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multi-modal maps) and for some multi-dimensional maps.

In this chapter, we will obtain new characterizations and proofs of the uniform ergodicity properties established in Chapter 15. We will consider a Markov kernel P as a linear operator on a set of probability measures endowed with a certain metric. An invariant probability measure is a fixed point of this operator, and therefore, a natural idea is to use a fixed-point theorem to prove convergence of the iterates of the kernel to the invariant distribution. To do so, in Section 18.1, we will first state and prove a version of the fixed-point theorem that suits our purposes. As appears in the fixed-point theorem, the main restriction of this method is that it can provide only geometric rates of convergence. These techniques will be again applied in Chapter 20, where we will be dealing with other metrics on the space of probability measures.

We give a general definition of the topological pressureP top (f, S) for continuous real valued functionsf: X→ℝ on transitive countable state Markov shifts (X, S). A variational principle holds for functions satisfying a mild distortion property. We introduce a new notion of Z-recurrent functions. Given any such functionf, we show a general method how to obtain tight sequences of invariant probability measures supported on periodic points such that a weak accumulation pointμ is an equilibrium state forf if and only if ef − dμ<∞. We discuss some conditions that ensure this integrability. As an application we obtain the Gauss measure as a weak limit of measures supported on periodic points.