We give two new proofs that the Sinai–Ruelle–Bowen (SRB) measure t↦μt of a C2 path ft of unimodal piecewise expanding C3 maps is differentiable at 0 if ft is tangent to the topological class of f0. The arguments are more conceptual than the original proof of Baladi and Smania [Linear response formula for piecewise expanding unimodal maps. Nonlinearity21 (2008), 677–711], but require proving Hölder continuity of the infinitesimal conjugacy α (a new result, of independent interest) and using spaces of bounded p-variation. The first new proof gives differentiability of higher order of ∫ ψ dμt if ft is smooth enough and stays in the topological class of f0 and if ψ is smooth enough (a new result). In addition, this proof does not require any information on the decomposition of the SRB measure into regular and singular terms, making it potentially amenable to extensions to higher dimensions. The second new proof allows us to recover the linear response formula (i.e. the formula for the derivative at 0) obtained by Baladi and Smania, by an argument more conceptual than the ‘brute force’ cancellation mechanism used by Baladi and Smania.

A set Λ is internally chain transitive if for any x,y∈Λ and ϵ>0 there is an ϵ-pseudo-orbit in Λ between x and y. In this paper we characterize all ω-limit sets in shifts of finite type by showing that, if Λ is a closed, strongly shift-invariant subset of a shift of finite type, X, then there is a point z∈X with ω(z)=Λ if and only if Λ is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet ℬ is the ω-limit set of some point in the full shift space over ℬ. We use similar techniques to prove that, for a tent map f, a closed, strongly f-invariant, internally chain transitive subset of the interval is the ω-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift space Z𝒢 (a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not the ω-limit set of any point in Z𝒢.

The purpose of the present paper is to provide a link between skew-product systems and linear dynamics. In particular, we give a criterion for skew-products of linear operators to be topologically transitive. This is then applied to certain families of linear operators including scalar multiples of the backward shift, backward unilateral weighted shifts, composition, translation and differentiation operators. We also prove the existence of common hypercyclic vectors for certain families of skew-product systems.

In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.

We introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension one and hyperbolic, corresponding to the unique complete metric of curvature −1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivization of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation ρ:π1(S)→GL(n,ℂ). We give conditions under which the foliated geodesic flow has a generic repeller–attractor statistical dynamics. That is, there are measures μ− and μ+ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to μ− and for positive time to μ+ (i.e. μ+ is the unique Sinaï, Ruelle and Bowen (SRB)-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behaviour of the leaves of the foliation.

We introduce a natural equivalence relation on the space ℋ0 of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen–Marcus theorem. Furthermore, if η is such a measure and G acts on a probability space (X,μ) by measure-preserving transformations then η×μ is virtually ergodic with respect to a natural equivalence relation on ℋ0×X. This is comparable to a special case of the Howe–Moore theorem. These results are applied to prove a new ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space.

Let Γ be a dense subgroup of a simply connected nilpotent Lie group G generated by a finite symmetric set S. We consider the n-ball Sn for the word metric induced by S on Γ. We show that Sn (with uniform measure) becomes equidistributed on G with respect to the Haar measure as n tends to infinity. We also prove the analogous result for random walk 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 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.

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.

In this paper we study the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches and a parabolic fixed point. It turns out that, in strong contrast with the hyperbolic case, the domain of the spectrum is unbounded and points of non-differentiability might exist. Moreover, the spectrum is not concave. We establish conditions that ensure the existence of inflection points. To the best of our knowledge this is the first time that conditions of this type have been given. We also study the thermodynamic formalism for such maps. We prove that the pressure function is real analytic in a certain interval and then becomes equal to zero. We also discuss the existence and uniqueness of equilibrium measures. In order to do so, we introduce a family of countable Markov shifts that can be thought of as a generalization of the renewal shift.

We first study the rate of growth of ergodic sums along a sequence (an) of times: SNf(x)=∑ n≤Nf(Tanx). We characterize the maximal rate of growth and identify a number of sequences such as an=2n, along which the maximal rate of growth is achieved. To point out though the general character of our techniques, we then turn to Khintchine’s strong uniform distribution conjecture that the averages (1/N)∑ n≤Nf(nx mod 1) converge pointwise to ∫ f for integrable functions f. We give a simple, intuitive counterexample and prove that, in fact, divergence occurs at the maximal rate.

D. Ruelle and I. Putnam have constructed three C*-algebras from the homoclinic and heteroclinic structure of a Smale space. This paper gives gives a complete description of these algebras when the Smale space is one of the generalized one-dimensional solenoids studied by R. Williams and I. Yi.

We construct ergodic exponential maps fλ(z)=λez such that the forward orbit of the origin is dense in .