Let $(M,\{\phi^t\})$ be a smooth (not necessarily transitive) Anosov flow without fixed points generated by a vector field $X(x)=(d/dt)|_{t=0}\phi^t(x)$ on a compact manifold M. Let $A:M\rightarrow\mathbb{R}$ be a globally Hölder function defined on M. Assume that $\int_0^T A\circ\phi^t(x)\,dt\geq0$ for any periodic orbit $\{\phi^t(x)\}_{t=0}^{t=T}$ of period T. Then there exists a Hölder function $V:M\rightarrow \mathbb{R}$, called a sub-action, smooth in the flow direction, such that

\[A(x)\geq L_XV(x),\quad\text{for all }x\in M\]

(where $L_XV=(d/dt)|_{t=0}V\circ\phi^t(x)$ denotes the Lie derivative of V). If A is $\mathcal{C}^r$ then LXV is $\mathcal{C}^r$ on any local center-stable manifold.

We study the dynamics of compositions of a sequence of holomorphic mappings in $\mathbb{P}^k$. We define ergodicity and mixing for non-autonomous dynamical systems, and we construct totally invariant measures for which our sequence satisfies these properties.

A $\lambda$-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. The present author has previously constructed a C*-algebra $\mathcal{ O}_{\mathfrak L}$ associated with a $\lambda$-graph system $\mathfrak L$ that is a generalization of the Cuntz–Krieger algebras. In this paper, we introduce an entropic quantity for the $\lambda$-graph system $\mathfrak L$, called the volume entropy for $\mathfrak L$ and written as $h_{\rm vol}(\mathfrak L)$. The volume entropy $h_{\rm vol}(\mathfrak L)$ is invariant under shift equivalence of $\lambda$-graph systems, and yields a new topological conjugacy invariant of subshifts. We prove that Voiculescu's non-commutative topological entropy for the canonical completely positive map of the C*-algebra $\mathcal{O}_{\mathfrak L}$ is the volume entropy $h_{\rm vol}(\mathfrak L)$.

An example of a two-dimensional subshift is the set of arrays $\{x_{(i,j)}:(i,j)\in\mathbb{Z}^2\}$ of zeros and ones for which $x_{(i,j)}+x_{(i+1,j)}+x_{(i,j+1)}=0$ for each $i,j\in\mathbb{Z}$, where addition is taken modulo 2. There are two shift operators, the horizontal shift and the vertical shift. This subshift is called the Ledrappier example or the three dot dynamical system. In this paper we give an internal description of all transitive subshifts for which one of the shift operators is sofic in the usual one-dimensional sense.

In this paper we show that a continuous map f from a connected graph G to itself is pointwise-recurrent if and only if one of the following two statements holds: (1) G is a circle, and f is a homeomorphism topologically conjugate to an irrational rotation of the unit circle S1; (2) f is a periodic homeomorphism.

We use a notion (introduced in Topology41 (2002), 1119–1212), which is stronger than the concept of filtration pair, to prove a stable/unstable manifold general theorem for local homeomorphisms of the plane in a neighborhood of an isolated fixed point.

Given a Lagrangian L, Mather introduced the $\beta$-function of L, which is a convex function. Many interesting properties of the Euler–Lagrange flow can be derived from the study of the behaviour of the $\beta$-function. In this work we obtain some links between the vertices of the $\beta$-function and the Hausdorff dimension of the associated invariant measures, from which we recover a result of differentiability of the $\beta$-function.

In this paper we use infinite ergodic theory to study limit sets of essentially free Kleinian groups which may have parabolic elements of arbitrary rank. By adapting a method of Adler, we construct a section map S for the geodesic flow on the associated hyperbolic manifold. We then show that this map has the Markov property and that it is conservative and ergodic with respect to the invariant measure induced by the Liouville–Patterson measure. Furthermore, we obtain that S is rationally ergodic with respect to different types of return sequences (an), which are governed by the exponent of convergence $\delta$ and the maximal possible rank kmax of the parabolic elements of the group as follows \[a_n\asymp\begin{cases}n^{2\delta-k_{\max}}&\text{for }\delta<(k_{\max}+1)/2,\\n/\!\log n&\text{for }\delta=(k_{\max}+1)/2,\\n&\text{for }\delta>(k_{\max}+1)/2.\end{cases}\] Subsequently, we give a discussion of an associated canonical map T, which is an analogue of the Bowen–Series map in the Fuchsian case. We show that T is pointwise dual ergodic with respect to these return sequences (an), which then allows us to determine the index of variation $\beta=\min\{1,2\delta-k_{\max}\}$, and to deduce that the ergodic sums Sn(f)/an converge strongly distributional to the Mittag–Leffler distribution of index $\beta$. We then give applications to number theory and to the statistics of cuspidal windings. Also, as a corollary we obtain a special case of Sullivan's result that the geodesic flow on a geometrically finite hyperbolic manifold is ergodic with respect to the Liouville–Patterson measure.

Let M be the four-dimensional compact manifold $M=T^2\times S^2$ and let $k\ge2$. We construct a $C^\infty$ diffeomorphism $F:M\to M$ with precisely k intermingled minimal attractors $A_1,\dotsc, A_k$. Moreover the union of the basins is a set of full Lebesgue measure. This means that Lebesgue almost every point in M lies in the basin of attraction of Aj for some j, but every non-empty open set in M has a positive measure intersection with each basin. We also construct $F:M\to M$ with a countable infinity of intermingled minimal attractors.

In this paper, we consider piecewise smooth dynamical systems with generalized indifferent periodic orbits associated to a given potential function. We show that the presence of such orbits causes non-uniqueness of equilibrium states (phase transitions) and non-Gibbsianness of equilibrium measures. If, in addition, the system exhibits local exponential instability, we show that the distribution of the stopping times over hyperbolic regions has powerlike tails with respect to any s-conformal measure, where s is the smallest zero of Bowen's equation. Finally, we prove that the Hausdorff dimension of the limit set is equal to s provided the dynamics is piecewise conformal.

This paper deals with families of conformal iterated function systems (CIFS). The space of all CIFS, with common seed space X and alphabet I, is successively endowed with the topology of pointwise convergence and a new, weaker topology called $\lambda$-topology. It is proved that the pressure and the Hausdorff dimension of the limit set are continuous with respect to the topology of pointwise convergence when I is finite, and are lower semi-continuous, though generally not continuous, when I is infinite. It is then shown that these two functions are, in any case, continuous in the $\lambda$-topology. The concepts of analytic, regularly analytic and plane-analytic families of CIFS are also introduced. It is established that if a family of CIFS is regularly analytic, then the Hausdorff dimension function is real-analytic; if a family is plane-analytic, then the Hausdorff dimension function is continuous and subharmonic, though not necessarily real-analytic. These results are then applied to finite parabolic CIFS. Counter-examples highlighting breakdowns of real-analyticity in the Hausdorff dimension among analytic, but not regularly analytic, families are further provided. Such families often exhibit a phenomenon known as phase transition. Sufficient conditions preventing the occurrence of such transitions are supplemented.

We describe a pair of invariants for actions of finite groups on shifts of finite type, the left-reduced and right-reduced shifts. The left-reduced shift was first constructed by Fiebig, who showed that its zeta function is an invariant, and in fact equal to the zeta function of the quotient dynamical system. We also give conditions for expansivity of the quotient, and applications to combinatorial group theory, knot theory and topological quantum field theory.

For countable-to-one transitive Markov maps, we show that the natural extensions of invariant ergodic weak Gibbs measures absolutely continuous with respect to weak Gibbs conformal measures possess a version of the u-Gibbs property. In particular, if dynamical potentials admit generalized indifferent periodic points then the natural extensions exhibit a non-Gibbsian character in statistical mechanics. Our results can be applicable to certain non-hyperbolic number-theoretical transformations of which natural extensions possess unstable (respectively stable) leaves with subexponential expansion (respectively contraction).

We prove a non-conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.