This paper considers two commuting smooth transformations on a Banach space and proves the sub-additivity of the measure theoretic entropies under mild conditions. Furthermore, some additional conditions are given for the equality of the entropies. This extends Hu’s work [Some ergodic properties of commuting diffeomorphisms. Ergod. Th. & Dynam. Sys. 13(1) (1993), 73–100] about commuting diffeomorphisms in a finite dimensional space to the case of systems on an infinite dimensional Banach space.

We introduce the mean topological dimension of random bundle transformations associated with an infinite countable discrete amenable group action and show that continuous bundle random dynamical systems for amenable groups with finite fibre topological entropy have zero mean topological dimensions.

This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in Mierczyński et al. [18]. Two different approaches to its existence are presented. The state space X will be a separable ordered Banach space with $\dim X\geq 2$, dual space $X^{*}$, and positive cone $X^+$ normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition $X^*$ is also separable, one obtains an exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds. Detailed examples are given for all the situations, including also a case where the cone is not normal.

We consider time-inhomogeneous ordinary differential equations (ODEs) whose parameters are governed by an underlying ergodic Markov process. When this underlying process is accelerated by a factor $\varepsilon^{-1}$, an averaging phenomenon occurs and the solution of the ODE converges to a deterministic ODE as $\varepsilon$ vanishes. We are interested in cases where this averaged flow is globally attracted to a point. In that case, the equilibrium distribution of the solution of the ODE converges to a Dirac mass at this point. We prove an asymptotic expansion in terms of $\varepsilon$ for this convergence, with a somewhat explicit formula for the first-order term. The results are applied in three contexts: linear Markov-modulated ODEs, randomized splitting schemes, and Lotka–Volterra models in a random environment. In particular, as a corollary, we prove the existence of two matrices whose convex combinations are all stable but are such that, for a suitable jump rate, the top Lyapunov exponent of a Markov-modulated linear ODE switching between these two matrices is positive.

In this paper, we study a connection between disintegration of measures and geometric properties of probability spaces. We prove a disintegration theorem, addressing disintegration from the perspective of an optimal transport problem. We look at the disintegration of transport plans, which are used to define and study disintegration maps. Using these objects, we study the regularity and absolute continuity of disintegration of measures. In particular, we exhibit conditions for which the disintegration map is weakly continuous and one can obtain a path of measures given by this map. We show a rigidity condition for the disintegration of measures to be given into absolutely continuous measures.

In this paper, we consider random iterations of polynomial maps $z^{2} + c_{n}$, where $c_{n}$ are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter $c = -1$, almost every random Julia set is totally disconnected with much smaller radial parameters r than expected. We also introduce several open questions worth discussing.

In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $\nu $. We consider a class of non-square-integrable observables $\phi $, mostly of form $\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$, where $x_0$ is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index $\alpha \in (0,2)$. The two types of maps we concatenate are a class of piecewise $C^2$ expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and $\alpha $, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by $\nu $. This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.

We consider orthogonally invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {C})$. Astérisque 287 (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

We study a class of ordinary differential equations with a non-Lipschitz point singularity that admits non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on a parameter $\nu $: the regularized dynamics is globally defined for each $\nu> 0$, and the original singular system is recovered in the limit of vanishing $\nu $. We prove that this limit yields a unique statistical solution independent of regularization when the deterministic system possesses a chaotic attractor having a physical measure with the convergence to equilibrium property. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.

In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu $ on M where the transition kernel ${\mathcal P}$ admits an eigenfunction $0\leq \eta \in L^1(M,\mu )$. We find conditions on the transition densities of ${\mathcal P}$ with respect to $\mu $ which ensure that $\eta (x) \mu (\mathrm {d} x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $\mu $-almost surely. We apply this result to the random logistic map $X_{n+1} = \omega _n X_n (1-X_n)$ absorbed at ${\mathbb R} \setminus [0,1],$ where $\omega _n$ is an independent and identically distributed sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$

We study stationary measures for iterated function systems (considered as random dynamical systems) consisting of two piecewise affine interval homeomorphisms, called Alsedà–Misiurewicz (AM) systems. We prove that for an open set of parameters, the unique non-atomic stationary measure for an AM system has Hausdorff dimension strictly smaller than $1$. In particular, we obtain singularity of these measures, answering partially a question of Alsedà and Misiurewicz [Random interval homeomorphisms. Publ. Mat. 58(suppl.) (2014), 15–36].

We first introduce the concept of weak random periodic solutions of random dynamical systems. Then, we discuss the existence of such periodic solutions. Further, we introduce the definition of weak random periodic measures and study their relationship with weak random periodic solutions. In particular, we establish the existence of invariant measures of random dynamical systems by virtue of their weak random periodic solutions. We use concrete examples to illustrate the weak random periodic phenomena of dynamical systems induced by random and stochastic differential equations.

Several authors have shown that Kusuoka’s measure κ on fractals is a scalar Gibbs measure; in particular, it maximizes a pressure. There is also a different approach, in which one defines a matrix-valued Gibbs measure µ, which induces both Kusuoka’s measure κ and Kusuoka’s bilinear form. In the first part of the paper, we show that one can define a ‘pressure’ for matrix-valued measures; this pressure is maximized by µ. In the second part, we use the matrix-valued Gibbs measure µ to count periodic orbits on fractals, weighted by their Lyapounov exponents.

In this paper, we show that, with probability $1$, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold’s 1965 speculation that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov–Sodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including Kolmogorov–Arnold–Moser (KAM) theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on ${\mathbb {R}}^3$ and of high-frequency Beltrami fields on the 3-torus.

We construct a geometrico-symbolic version of the natural extension of the random $\beta $-transformation introduced by Dajani and Kraaikamp [Random $\beta $-expansions. Ergod. Th. & Dynam. Sys. 23(2) (2003) 461–479]. This construction provides a new proof of the existence of a unique absolutely continuous invariant probability measure for the random $\beta $-transformation, and an expression for its density. We then prove that this natural extension is a Bernoulli automorphism, generalizing to the random case the result of Smorodinsky [$\beta $-automorphisms are Bernoulli shifts. Acta Math. Acad. Sci. Hungar. 24 (1973), 273–278] about the greedy transformation.

We prove a random Ruelle–Perron–Frobenius theorem and the existence of relative equilibrium states for a class of random open and closed interval maps, without imposing transitivity requirements, such as mixing and covering conditions, which are prevalent in the literature. This theorem provides the existence and uniqueness of random conformal and invariant measures with exponential decay of correlations, and allows us to expand the class of examples of (random) dynamical systems amenable to multiplicative ergodic theory and the thermodynamic formalism. Applications include open and closed non-transitive random maps, and a connection between Lyapunov exponents and escape rates through random holes. We are also able to treat random intermittent maps with geometric potentials.

We find generalized conformal measures and equilibrium states for random dynamics generated by Ruelle expanding maps, under which the dynamics exhibits exponential decay of correlations. This extends results by Baladi [Correlation spectrum of quenched and annealed equilibrium states for random expanding maps. Comm. Math. Phys. 186 (1997), 671–700] and Carvalho et al [Semigroup actions of expanding maps. J. Stat. Phys. 116(1) (2017), 114–136], where the randomness is driven by an independent and identically distributed process and the phase space is assumed to be compact. We give applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and introduce a boundary of equilibria for not necessarily free semigroup actions.

For $k \geq 2$, we prove that in a $C^{1}$-open and $C^{k}$-dense set of some classes of $C^{k}$-Anosov flows, all Lyapunov exponents have multiplicity one with respect to appropriate measures. The classes are geodesic flows with equilibrium states of Holder-continuous potentials, volume-preserving flows, and all fiber-bunched Anosov flows with equilibrium states of Holder-continuous potentials.

In this article, we continue the structural study of factor maps between symbolic dynamical systems and the relative thermodynamic formalism. Here, one is studying a factor map from a shift of finite type X (equipped with a potential function) to a sofic shift Z, equipped with a shift-invariant measure $\nu $. We study relative equilibrium states, that is, shift-invariant measures on X that push forward under the factor map to $\nu $ which maximize the relative pressure: the relative entropy plus the integral of $\phi $. In this paper, we establish a new connection to multiplicative ergodic theory by relating these factor triples to a cocycle of Ruelle–Perron–Frobenius operators, and showing that the principal Lyapunov exponent of this cocycle is the relative pressure; and the dimension of the leading Oseledets space is equal to the number of measures of relative maximal entropy, counted with a previously identified concept of multiplicity.

We consider families of random products of close-by Anosov diffeomorphisms, and show that statistical stability and linear response hold for the associated families of equivariant and stationary measures. Our analysis relies on the study of the top Oseledets space of a parametrized transfer operator cocycle, as well as ad-hoc abstract perturbation statements. As an application, we show that, when the quenched central limit theorem (CLT) holds, under the conditions that ensure linear response for our cocycle, the variance in the CLT depends differentiably on the parameter.