In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every invertible ergodic dynamical system, for every increasing sequence (an)n∈ℕ⊂ℝ+ such that an↗∞ and an/n→0 as n→∞, there exists a dense Gδ of measurable sets A such that the sequence of the distributions of the partial sums is dense in the set of the probability measures on ℝ.

We construct an infinite sequence of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of Abelian differentials. The existence of such a sequence shows that all pseudo-Anosov homeomorphisms fixing orientable measured foliations cannot be obtained by the Rauzy–Veech induction strategy.

The Ruelle operator theorem has been studied extensively both in dynamical systems and iterated function systems. In this paper we study the Ruelle operator theorem for non-expansive systems. Our theorems give some sufficient conditions for the Ruelle operator theorem to be held for a non-expansive system.

A basic nilsequence is a sequence of the form ψ(n)=f(Tnx), where x is a point of a compact nilmanifold X, T is a translation on X, and f∈C(X); a nilsequence is a uniform limit of basic nilsequences. Let X=G/Γ be a compact nilmanifold, Y be a subnilmanifold of X, g(n) be a polynomial sequence in G, and f∈C(X); we show that the sequence ∫ g(n)Yf, n∈ℤ, is the sum of a basic nilsequence and a sequence that converges to zero in uniform density. This implies that, given an ergodic invertible measure-preserving system (W,ℬ,μ,T), with μ(W)<∞, polynomials p1,…,pk∈ℤ[n], and sets A1,…,Ak∈ℬ, the sequence μ(Tp1(n)A1∩⋯∩Tpk(n)Ak) is the sum of a nilsequence and a sequence that converges to zero in uniform density. We also obtain a version of this result for the case where pi are polynomials in several variables.

A sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure-preserving system (X,ℬ,μ,T) and any bounded measurable function f, the averages (1/N)∑ Nn=1f(Tsnx) converge in the L2(μ) norm. We construct a sequence (sn) which is good for the mean ergodic theorem but such that the sequence (s2n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (skn) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form (1/N)∑ Nn=1f1(Tsnx)f2(T2snx)⋯fℓ(Tℓsnx). We also prove a similar result for pointwise convergence of single ergodic averages.

We prove that if a Borel probability measure on the circle group is invariant under the action of a ‘large’ multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then the measure is either Lebesgue or has finite support.

We introduce a renormalization group scheme that applies to vector fields on 𝕋d×ℝm with frequency vectors that satisfy a Brjuno condition. Earlier approaches were restricted to Diophantine frequencies, owing to a limited control of multidimensional continued fractions. We get around this restriction by avoiding the use of a continued-fractions expansion. Our results concerning invariant tori generalize those of [H. Koch and S. Kocić, Renormalization of vector fields and Diophantine invariant tori. Ergod. Th. & Dynam. Sys.28 (2008), 1559–1585] from Diophantine- to Brjuno-type frequency vectors. In particular, each Brjuno vector ω∈ℝd determines an analytic manifold 𝒲 of infinitely renormalizable vector fields, and each vector field on 𝒲 is shown to have an elliptic invariant d-torus with frequencies ω1,ω2,…,ωd.

We study the dynamics of a type of non-conservative billiards where the ball is ‘kicked’ by the wall giving a new impulse in the direction of the normal. For different types of billiard tables we study the existence of attractors with dominated splitting.

The purpose of this paper is to exhibit highly structured subdynamics for a class of non-expansive algebraic ℤd-actions based on the closed orbits of elements of an action. This is done using dynamical Dirichlet series to encode orbit counts. It is shown that there is a distinguished group homomorphism from ℤd onto a finite abelian group that controls the form of the Dirichlet series of elements of an action and that these series have common analytic properties. Corresponding orbit growth asymptotics are subsequently investigated.

We consider the Rényi entropy function for weakly ψ-mixing systems. The first main result of the paper establishes existence and regularity properties. The second main result is obtaining the decay rate for the large deviation of the return time to cylinder sets; we show that it is exponential with a rate given by the Rényi entropy function. Finally, we also obtain bounds for the free energy.

We consider a class of ordinary differential equations describing one-dimensional systems with a quasi-periodic forcing term and in the presence of large damping. We discuss the conditions to be assumed on the mechanical force and the forcing term for the existence of quasi-periodic solutions which have the same frequency vector as the forcing.

We propose a generalization of Exel’s crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler’s theory of Toeplitz and Cuntz–Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz–Pimsner algebras of product systems.

We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets goes to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.

We consider a subclass of tilings: the tilings obtained by cut-and-projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent α in terms of the ranks of certain groups which appear in the construction. We give bounds for α. These computations apply to some well-known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over ℚ. We show that such a link cannot be established for more general tilings, and we present a counterexample in dimension one.

Let K be a compact metrizable group and Γ be a finitely generated group of commuting automorphisms of K. We show that ergodicity of Γ implies Γ contains ergodic automorphisms if center of the action, Z(Γ)={α∈Aut(K)∣α commutes with elements of Γ} has descending chain condition. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which, in particular, provide new proofs to the theorems of Berend [Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc.289(1) (1985), 393–407] and Schmidt [Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480–496].

We study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of topological generators (elements generating a dense subgroup) of full groups allowing us to distinguish between full groups of equivalence relations generated by free, ergodic actions of the free groups Fm and Fn if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group if an only if its full group is topologically finitely generated.

Let ℳ={f(z)=(zm/sinh m z) for z∈ℂ∣ either m or m/2 is an odd natural number}. For each f∈ℳ, the set of singularities of the inverse function of f is an unbounded subset of the real line ℝ. In this paper, the iteration of functions in one-parameter family 𝒮={fλ(z)=λf(z)∣λ∈ℝ∖{0}} is investigated for each f∈ℳ. It is shown that, for each f∈ℳ, there is a critical parameter λ*>0 depending on f such that a period-doubling bifurcation occurs in the dynamics of functions fλ in 𝒮 when the parameter |λ| passes through λ*. The non-existence of Baker domains and wandering domains in the Fatou set of fλ is proved. Further, it is shown that the Fatou set of fλ is infinitely connected for 0<∣λ∣≤λ* whereas for ∣λ∣≥λ*, the Fatou set of fλ consists of infinitely many components and each component is simply connected.

Let (X,T) be a topological dynamical system (TDS), and h(T,K) the topological entropy of a subset K of X. (X,T) is lowerable if for each 0≤h≤h(T,X) there is a non-empty compact subset with entropy h; it is hereditarily lowerable if each non-empty compact subset is lowerable; it is hereditarily uniformly lowerable if for each non-empty compact subset K and each 0≤h≤h(T,K) there is a non-empty compact subset Kh⊆K with h(T,Kh)=h and Kh has at most one limit point. It is shown that each TDS with finite entropy is lowerable, and that a TDS (X,T) is hereditarily uniformly lowerable if and only if it is asymptotically h-expansive.

Let the invariant probability measures for an orientation-reversing weakly expanding map of the interval [0,1] be partially ordered by majorization. The minimal elements of the resulting poset are shown to be convex combinations of Dirac measures supported on two adjacent fixed points. A consequence is that if f:[0,1]→ℝ is strictly convex, then either its minimizing measure is unique and is a Dirac measure on a fixed point, or f has precisely two ergodic minimizing measures, namely Dirac measures on two adjacent fixed points. In the case where {0,1} is a period-two orbit, with corresponding invariant measure μ01, the maximal elements of the poset are shown to be convex combinations of μ01 with the Dirac measure on either the leftmost, or the rightmost, fixed point. This facilitates the identification of f-maximizing measures when f:[0,1]→ℝ is convex.

We show that a Poisson structure whose linear part vanishes can be holomorphically normalized in a neighbourhood of its singular point if, on the one hand, a Diophantine condition on a Lie algebra associated to the quadratic part is satisfied and, on the other hand, the normal form satisfies some formal conditions.