A dynamical 2-complex is a 2-complex equipped with a set of combinatorial properties which allow one to define non-singular semi-flows on the complex. After giving a combinatorial characterization of the dynamical 2-complexes which define hyperbolic attractors when embedded in compact 3-manifolds, we give an effective criterion for the existence of cross sections to the semi-flows on these 2-complexes. In the embedded case, this gives an effective criterion of existence of cross sections to the associated hyperbolic attractors. We present a similar criterion for boundary-tangent flows on compact 3-manifolds which are constructed by means of our dynamical 2-complexes.

Let $M$ be an analytic connected 2-manifold with empty boundary, over the ground field $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. Let $Y$ and $X$ denote differentiable vector fields on $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f:\,M\rightarrow \mathbb{F}$. A subset $K$ of the zero set $\mathsf{Z}(X)$ is an essential block for $X$ if it is non-empty, compact and open in $\mathsf{Z}(X)$, and the Poincaré–Hopf index $\mathsf{i}_{K}(X)$ is non-zero. Let ${\mathcal{G}}$ be a finite-dimensional Lie algebra of analytic vector fields that tracks a non-trivial analytic vector field $X$. Let $K\subset \mathsf{Z}(X)$ be an essential block. Assume that if $M$ is complex and $\mathsf{i}_{K}(X)$ is a positive even integer, no quotient of ${\mathcal{G}}$ is isomorphic to $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. Then ${\mathcal{G}}$ has a zero in $K$ (main result). As a consequence, if $X$ and $Y$ are analytic, $X$ is non-trivial, and $Y$ tracks $X$, then every essential component of $\mathsf{Z}(X)$ meets $\mathsf{Z}(Y)$. Fixed-point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

We consider the motion of a classical particle under the influence of a random potential on ℝd, in particular the distribution of asymptotic velocities and the question of ergodicity of time evolution.

We give a correction to the assumption of Theorems 1.12 and 2.6 in the paper [H. Sumi. Semi-hyperbolic fibered rational maps and rational semigroups. Ergod. Th. & Dynam. Sys.26 (2006), 893–922].

Except for the painful beginning and end, well-deserved good fortune accompanied Moser all his life. Accomplishment and recognition came to him early and lastingly. He was the first recipient of the G. D. Birkhoff Prize of the AMS and SIAM. He was honored by a Wolf Prize, medals were showered on him, academies vied to elect him as their member. He was an outstanding expositor, much in demand. He delivered the Gibbs lecture of the AMS, the Pauli lectures of the ETH, the Hardy lectures of Cambridge University, the Hermann Weyl lectures of the Accademia Lincei, and the von Neumann lecture of SIAM. He was president of the IMU for three years.

He was fortunate in having many good students, with whom he continued close scientific and personal relations. He loved his family fiercely, two daughters and a stepson, six grandchildren and charming and talented sons-in-law.

According to an old Roman proverb, ‘what doesn't destroy me makes me stronger’. This fits Jürgen, for in his early youth he was exposed to deadly danger, but survived the stronger for it. In 1938, the ten year old Jürgen was offered a chance to enroll in an elite boarding school for future leaders, but his parents who saw the Nazi's plan were able to extricate him. Jürgen's father was a neurologist, and as such was sometimes called upon to declare someone as ‘unfit to live’ according to the Nazi eugenics program. Jürgen's father, courageously, always refused to issue such certifications.

We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism L of $\mathbb T^2$ . It is known that the positive Lyapunov exponent of f with respect to the normalized Lebesgue measure is less than or equal to the topological entropy of L, which, in addition, is less than or equal to the Lyapunov exponent of f with respect to the probability measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on these two dynamical invariants.

We show that there is a measure-preserving system $(X,\mathscr {B}, \mu , T)$ together with functions $F_0, F_1, F_2 \in L^{\infty }(\mu )$ such that the correlation sequence $C_{F_0, F_1, F_2}(n) = \int _X F_0 \cdot T^n F_1 \cdot T^{2n} F_2 \, d\mu $ is not an approximate integral combination of $2$ -step nilsequences.

Given a factor code between sofic shifts X and Y, there is a family of decompositions of the original code into factor codes such that the entropies of the intermediate subshifts arising from the decompositions are dense in the interval from the entropy of Y to that of X. Furthermore, if X is of finite type, we can choose those intermediate subshifts as shifts of finite type. In the second part of the paper, given an embedding from a shift space to an irreducible sofic shift, we characterize the set of the entropies of the intermediate subshifts arising from the decompositions of the given embedding into embeddings.

Let X be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank-one axis. Assume X is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of oriented closed geodesics of length at most t; then $\lim \nolimits _{t \to \infty } P_t / ({e^{ht}}/{ht}) = 1$ , where h is the entropy of the geodesic flow on the space $GX$ of parametrized unit-speed geodesics in X.

A classical approach to the study of a dynamical system is to code it using a partition. This leads to the question of the quality of this coding and in particular to the problem of finding checkable conditions ensuring that the coding is essentially one-to-one. We prove that, with respect to an invariant measure with only positive Lyapunov exponents, it is basically enough that the map be one-to-one on each piece of the partition. This is a multi-dimensional generalization of a well-known fact in one-dimensional dynamics. We apply this result to the study of the measure of maximal entropy of endomorphisms of complex projective spaces.

We consider piecewise $C^{2}$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^{2}$ non-flat map of the interval displays only a finite number of non-periodic attractors.

For $\mathscr {B} \subseteq \mathbb {N} $, the $ \mathscr {B} $-free subshift $ X_{\eta } $ is the orbit closure of the characteristic function of the set of $ \mathscr {B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_{\eta } $, have their analogues for $ X_{\eta } $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures [G. Keller. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21(3) (2021), Paper No. 2140008]. A central assumption in our work is that $\eta ^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_{\eta } $) is regular. From this, we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_{\eta } $ from above and below.

The cohomology action of an Anosov diffeomorphism on a nilmanifold resembles that of a Cartesian product map. Corresponding results hold for infranilmanifolds, giving an invariant bigrading of the cohomology and a fourfold symmetry that extends Poincaré duality. Holonomy invariant cocycles are applied to the action on first cohomology.

We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities.

Let G be the group $\text {PAff}_{+}(\mathbb R/\mathbb Z)$ of piecewise affine circle homeomorphisms or the group ${\operatorname {\mathrm {Diff}}}^{{\kern1pt}\infty }(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in G is given.

If K is a countable amenable group acting freely and ergodically on a probability space (Γ, μ), and G is an arbitrary countable amenable group, we construct an injection of the space of unitary representations of G into the space of unitary 1-cocyles for K on (Γ, μ); this injection preserves intertwining operators. We apply this to show that for many of the standard non-type-I amenable groups H, the representation theory of H contains that of every countable amenable group.

In this paper, we reduce the logarithmic Sarnak conjecture to the $\{0,1\}$-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the $1$-Fourier uniformity conjecture, where the frequencies are restricted to any subset of $[0,1]$ with packing dimension less than one.

In this paper, we investigate pigeonhole statistics for the fractional parts of the sequence $\sqrt {n}$. Namely, we partition the unit circle $ \mathbb {T} = \mathbb {R}/\mathbb {Z}$ into N intervals and show that the proportion of intervals containing exactly j points of the sequence $(\sqrt {n} + \mathbb {Z})_{n=1}^N$ converges in the limit as $N \to \infty $. More generally, we investigate how the limiting distribution of the first $sN$ points of the sequence varies with the parameter $s \geq 0$. A natural way to examine this is via point processes—random measures on $[0,\infty )$ which represent the arrival times of the points of our sequence to a random interval from our partition. We show that the sequence of point processes we obtain converges in distribution and give an explicit description of the limiting process in terms of random affine unimodular lattices. Our work uses ergodic theory in the space of affine unimodular lattices, building upon work of Elkies and McMullen [Gaps in $\sqrt {n}$ mod 1 and ergodic theory. Duke Math. J. 123 (2004), 95–139]. We prove a generalisation of equidistribution of rational points on expanding horocycles in the modular surface, working instead on nonlinear horocycle sections.

Let f:Tn→Tn (n ≥ 3) be a hyperbolic toral automorphism. Let A be the set of α > 0 such that there is a Hölder continuous path of index α in Tn with 1-dimensional orbit-closure under f We prove that α0 = sup A can be expressed in terms of the eigenvalues of f and that α0 ∈ A if and only if α0 < 1.