Consider the quadratic family $T_a(x) = a x (1 - x)$ for $x \in [0, 1]$ and mixing Collet–Eckmann (CE) parameters $a \in (2,4)$. For bounded $\varphi $, set $\tilde \varphi _{a} := \varphi - \int \varphi \, d\mu _a$, with $\mu _a$ the unique acim of $T_a$, and put $(\sigma _a (\varphi ))^2 := \int \tilde \varphi _{a}^2 \, d\mu _a + 2 \sum _{i>0} \int \tilde \varphi _{a} (\tilde \varphi _{a} \circ T^i_{a}) \, d\mu _a$. For any mixing Misiurewicz parameter $a_{*}$, we find a positive measure set $\Omega _{*}$ of mixing CE parameters, containing $a_{*}$ as a Lebesgue density point, such that for any Hölder $\varphi $ with $\sigma _{a_{*}}(\varphi )\ne 0$, there exists $\epsilon _\varphi>0$ such that, for normalized Lebesgue measure on $\Omega _{*}\cap [a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$, the functions $\xi _i(a)=\tilde \varphi _a(T_a^{i+1}(1/2))/\sigma _a (\varphi )$ satisfy an almost sure invariance principle (ASIP) for any error exponent $\gamma>2/5$. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann’s proof for piecewise expanding maps. We need to introduce a variant of Benedicks–Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from Baladi et al [Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201 (2015), 773–844].

The first part of this work is devoted to the study of higher derivatives of pressure functions of Hölder potentials on shift spaces with finitely many symbols. By describing the derivatives of pressure functions via the central limit theorem for the associated random processes, we discover some rigid relationships between derivatives of various orders. The rigidity imposes obstructions on fitting candidate convex analytic functions by pressure functions of Hölder potentials globally, which answers a question of Kucherenko and Quas. In the second part of the work, we consider fitting candidate analytic germs by pressure functions of locally constant potentials. We prove that all 1-level candidate germs can be realised by pressures of some locally constant potentials, as long as the number of symbols in the symbolic set is large enough. There are also some results on fitting 2-level germs by pressures of locally constant potentials obtained in the work.

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 show that stationary time series can be uniformly approximated over all finite time intervals by mixing, non-ergodic, non-mean-ergodic, and periodic processes, and by codings of aperiodic processes. A corollary is that the ergodic hypothesis—that time averages will converge to their statistical counterparts—and several adjacent hypotheses are not testable in the non-parametric case. Further Baire category implications are also explored.

In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.

The present paper deals with the kinetic-theoretic description of the evolution of systems consisting of many particles interacting not only with each other but also with the external world, so that the equation governing their evolution contains an additional term representing such interaction, called the ‘forcing term’. Firstly, the interactions between pairs of particles are both conservative and nonconservative; the latter represents, among others, birth/death rates. The ‘forcing term’ does not express a ‘classical’ force exerted by the external world on the particles, but a more general influence on the effects of mutual interactions of particles, for instance, climate changes, that increase or decrease the different agricultural productions at different times, thus altering the economic relationships between different subsystems, that in turn can be also perturbed by stock market fluctuations, sudden wars, periodic epidemics, and so on. Thus, the interest towards these problems moves the mathematical analysis of the effects of different kinds of forcing terms on solutions to equations governing the collective (that is statistical) behaviour of such nonconservative many-particle systems. In the present paper, we offer a study of the basic mathematical properties of such solutions, along with some numerical simulations to show the effects of forcing terms for a classical prey–predator model in ecology.

Given a dynamical system, we prove that the shortest distance between two n-orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math. 344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit.

In this paper, we consider the convergence rate with respect to Wasserstein distance in the invariance principle for deterministic non-uniformly hyperbolic systems. Our results apply to uniformly hyperbolic systems and large classes of non-uniformly hyperbolic systems including intermittent maps, Viana maps, unimodal maps and others. Furthermore, as a non-trivial application to the homogenization problem, we investigate the Wasserstein convergence rate of a fast–slow discrete deterministic system to a stochastic differential equation.

Given a locally finite graph $\Gamma $, an amenable subgroup G of graph automorphisms acting freely and almost transitively on its vertices, and a G-invariant activity function $\unicode{x3bb} $, consider the free energy $f_G(\Gamma ,\unicode{x3bb} )$ of the hardcore model defined on the set of independent sets in $\Gamma $ weighted by $\unicode{x3bb} $. Under the assumption that G is finitely generated and its word problem can be solved in exponential time, we define suitable ensembles of hardcore models and prove the following: if $\|\unicode{x3bb} \|_\infty < \unicode{x3bb} _c(\Delta )$, there exists a randomized $\epsilon $-additive approximation scheme for $f_G(\Gamma ,\unicode{x3bb} )$ that runs in time $\mathrm {poly}((1+\epsilon ^{-1})\lvert \Gamma /G \rvert )$, where $\unicode{x3bb} _c(\Delta )$ denotes the critical activity on the $\Delta $-regular tree. In addition, if G has a finite index linearly ordered subgroup such that its algebraic past can be decided in exponential time, we show that the algorithm can be chosen to be deterministic. However, we observe that if $\|\unicode{x3bb} \|_\infty> \unicode{x3bb} _c(\Delta )$, there is no efficient approximation scheme, unless $\mathrm {NP} = \mathrm {RP}$. This recovers the computational phase transition for the partition function of the hardcore model on finite graphs and provides an extension to the infinite setting. As an application in symbolic dynamics, we use these results to develop efficient approximation algorithms for the topological entropy of subshifts of finite type with enough safe symbols, we obtain a representation formula of pressure in terms of random trees of self-avoiding walks, and we provide new conditions for the uniqueness of the measure of maximal entropy based on the connective constant of a particular associated graph.

Let $({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$. Consider the Birkhoff sums $S_n f = \sum _{k=0}^{n-1} f \circ T^{k}$, $n\geqslant 1$. For any $t \in {\mathbb R}$, denote by $\tau _t^f$ the first time when the sum $t+ S_n f$ leaves the positive half-line for some $n\geqslant 1$. By analogy with the case of random walks with independent and identically distributed increments, we study the asymptotic as $ n\to \infty $ of the probabilities $ \nu (x\in {\mathbb X}: \tau _t^f(x)>n) $ and $ {\nu (x\in {\mathbb X}: \tau _t^f(x)=n) }$. We also establish integral and local-type limit theorems for the sum $t+ S_n f(x)$ conditioned on the set $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$

We give an example of an FIID vertex-labeling of ${\mathbb T}_3$ whose marginals are uniform on $[0,1]$, and if we delete the edges between those vertices whose labels are different, then some of the remaining clusters are infinite. We also show that no such process can be finitary.

We use Gaussian measure-preserving systems to prove the existence and genericity of Lebesgue measure-preserving transformations $T:[0,1]\rightarrow [0,1]$ which exhibit both mixing and rigidity behavior along families of asymptotically linearly independent sequences. Let $\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _N\in [0,1]$ and let $\phi _1,\ldots ,\phi _N:\mathbb N\rightarrow \mathbb Z$ be asymptotically linearly independent (that is, for any $(a_1,\ldots ,a_N)\in \mathbb Z^N\setminus \{\vec 0\}$, $\lim _{k\rightarrow \infty }|\sum _{j=1}^Na_j\phi _j(k)|=\infty $). Then the class of invertible Lebesgue measure-preserving transformations $T:[0,1]\rightarrow [0,1]$ for which there exists a sequence $(n_k)_{k\in \mathbb {N}}$ in $\mathbb {N}$ with for any measurable $A,B\subseteq [0,1]$ and any $j\in \{1,\ldots ,N\}$, is generic. This result is a refinement of a result due to Stëpin (Theorem 2 in [Spectral properties of generic dynamical systems. Math. USSR-Izv. 29(1) (1987), 159–192]) and a generalization of a result due to Bergelson, Kasjan, and Lemańczyk (Corollary F in [Polynomial actions of unitary operators and idempotent ultrafilters. Preprint, 2014, arXiv:1401.7869]).

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.

Let $(X_k)_{k\geq 0}$ be a stationary and ergodic process with joint distribution $\mu $ , where the random variables $X_k$ take values in a finite set $\mathcal {A}$ . Let $R_n$ be the first time this process repeats its first n symbols of output. It is well known that $({1}/{n})\log R_n$ converges almost surely to the entropy of the process. Refined properties of $R_n$ (large deviations, multifractality, etc) are encoded in the return-time $L^q$ -spectrum defined as

provided the limit exists. We consider the case where $(X_k)_{k\geq 0}$ is distributed according to the equilibrium state of a potential with summable variation, and we prove that

where $P((1-q)\varphi )$ is the topological pressure of $(1-q)\varphi $ , the supremum is taken over all shift-invariant measures, and $q_\varphi ^*$ is the unique solution of $P((1-q)\varphi ) =\sup _\eta \int \varphi \,d\eta $ . Unexpectedly, this spectrum does not coincide with the $L^q$ -spectrum of $\mu _\varphi $ , which is $P((1-q)\varphi )$ , and it does not coincide with the waiting-time $L^q$ -spectrum in general. In fact, the return-time $L^q$ -spectrum coincides with the waiting-time $L^q$ -spectrum if and only if the equilibrium state of $\varphi $ is the measure of maximal entropy. As a by-product, we also improve the large deviation asymptotics of $({1}/{n})\log R_n$ .

Kingman’s subadditive ergodic theorem is traditionally proved in the setting of a measure-preserving invertible transformation T of a measure space $(X, \mu )$ . We use a theorem of Silva and Thieullen to extend the theorem to the setting of a not necessarily invertible transformation, which is non-singular under the assumption that $\mu $ and $\mu \circ T$ have the same null sets. Using this, we are able to produce versions of the Furstenberg–Kesten theorem and the Oseledeč ergodic theorem for products of random matrices without the assumption that the transformation is either invertible or measure-preserving.

We show that a piecewise monotonic map with positive topological entropy satisfies the level-2 large deviation principle with respect to the unique measure of maximal entropy under the conditions that the corresponding Markov diagram is irreducible and that the periodic measures of the map are dense in the set of ergodic measures. This result can apply to a broad class of piecewise monotonic maps, such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces.

In this paper we prove a local exponential synchronization for Markovian random iterations of homeomorphisms of the circle $S^{1}$ , providing a new result on stochastic circle dynamics even for $C^1$ -diffeomorphisms. This result is obtained by combining an invariance principle for stationary random iterations of homeomorphisms of the circle with a Krylov–Bogolyubov-type result for homogeneous Markov chains.

We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

It is shown that for a dense $G_\delta $ -subset of the subgroup of non-singular transformations (of a standard infinite $\sigma $ -finite measure space) whose Poisson suspensions are non-singular, the corresponding Poisson suspensions are ergodic and of Krieger’s type III1.

We show that the dynamical degree of an (i.i.d) random sequence of dominant, rational self-maps on projective space is almost surely constant. We then apply this result to height growth and height counting problems in random orbits.