It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is  $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.

Let $X$ be a solenoid, i.e. a compact, finite-dimensional, connected abelian group with normalized Haar measure $\unicode[STIX]{x1D707}$, and let $\unicode[STIX]{x1D6E4}\rightarrow \operatorname{Aff}(X)$ be an action of a countable discrete group $\unicode[STIX]{x1D6E4}$ by continuous affine transformations of $X$. We show that the probability measure preserving action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ does not have the spectral gap property if and only if there exists a $p_{\text{a}}(\unicode[STIX]{x1D6E4})$-invariant proper subsolenoid $Y$ of $X$ such that the image of $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually solvable group, where $p_{\text{a}}:\operatorname{Aff}(X)\rightarrow \operatorname{Aut}(X)$ is the canonical projection. When $\unicode[STIX]{x1D6E4}$ is finitely generated or when $X$ is the $a$-adic solenoid for an integer $a\geq 1$, the subsolenoid $Y$ can be chosen so that the image $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually abelian group. In particular, an action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ by affine transformations on a solenoid $X$ has the spectral gap property if and only if $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is strongly ergodic.

We formulate an abstract fluctuation theorem which sheds light on mathematical relations between the fluctuation theorems of Bochkov and Kuzovlev [Contribution to the general theory of thermal fluctuations in nonlinear systems. Sov. Phys.–JETP45 (1977), 125] and Jarzynski [Hamiltonian derivation of a detailed fluctuation theorem. J. Stat. Phys.98 (2001), 77–102] on the one hand, and those of Evans and Searles [Equilibrium microstates which generate second law violating steady states. Phys. Rev. E 50 (1994), 1645–1648] and Gallavotti and Cohen [Dynamical ensembles in stationary states. J. Stat. Phys.80 (1995), 931–970] on the other.

We prove that special flows over an ergodic rotation of the circle under a $C^{1}$ roof function with one discontinuity do not have local rank one. In particular, any such flow has infinite rank.

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 study invariant measures for a continuous function which maps a real interval into itself. We show that the ratio of the measures of the two subintervals into which it is divided by a fixed point is constrained by the the set of periods of periodic points. As a consequence of this we give a new forcing relation between periodic points.

We compute the Poisson boundary of locally discrete groups of diffeomorphisms of the circle.

For a piecewise monotonic map T:X\to{\Bbb R}, where X is a finite union of closed intervals, define R(T)= \bigcap_{n=0}^{\infty}\overline{T^{-n}X}. The influence of small perturbations of T on the dynamical system (R(T),T) is investigated. If P is a finite and T-invariant subset of R(T), and if f_0:P\to{\Bbb R} is a non-negative continuous function, then it is proved that the infimum of the topological pressure p(R(T),T,f) over all non-negative continuous functions f:X\to{\Bbb R} with f|_P=f_0 equals the maximum of h_{\text{\rm top}}(R(T),T) and p(P,T,f_0). This result is used to obtain stability conditions, which are equivalent to the upper semi-continuity of the topological pressure for every continuous function f:X\to{\Bbb R}. In the case of a continuous piecewise monotonic map T:X\to{\Bbb R} one of these stability conditions is: there exists no endpoint of an interval of monotonicity of T which is periodic and contained in the interior of X. Furthermore, these results are applied to monotonic mod one transformations, another special case of piecewise monotonic maps.

We show that a $C^{1}$-generic non-partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restricted to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^{1}$-generic set of symplectic diffeomorphisms far from partial hyperbolicity.

Given an $\unicode[STIX]{x1D6FC}>1$ and a $\unicode[STIX]{x1D703}$ with unbounded continued fraction entries, we characterize new relations between Sturmian subshifts with slope $\unicode[STIX]{x1D703}$ with respect to (i) an $\unicode[STIX]{x1D6FC}$-Hölder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of $\unicode[STIX]{x1D703}$, and (iii) complexity notions which we call $\unicode[STIX]{x1D6FC}$-repetitiveness, $\unicode[STIX]{x1D6FC}$-repulsiveness and $\unicode[STIX]{x1D6FC}$-finiteness—generalizations of the properties known as linear repetitiveness, repulsiveness and power freeness, respectively. We show that the level sets relate naturally to (exact) Jarník sets and prove that their Hausdorff dimension is $2/(\unicode[STIX]{x1D6FC}+1)$.

In this paper the notion of sub-exponential measure complexity for an invariant Borel probability measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential measure complexity is constructed.

Let $X$ be a compact, metric and totally disconnected space and let $f:X\rightarrow X$ be a continuous map. We relate the eigenvalues of $f_{\ast }:\check{H}_{0}(X;\mathbb{C})\rightarrow \check{H}_{0}(X;\mathbb{C})$ to dynamical properties of $f$, roughly showing that if the dynamics is complicated then every complex number of modulus different from 0, 1 is an eigenvalue. This stands in contrast with a classical inequality of Manning that bounds the entropy of $f$ below by the spectral radius of $f_{\ast }$.

We investigate the properties of absolutely continuous invariant probability measures (ACIPs), especially those measures with bounded variation densities, for piecewise area preserving maps (PAPs) on ℝd. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. Using a functional analytic approach, we first explore the relationship between topological transitivity and uniqueness of ACIPs, and then give an approach to construct invariant measures with bounded variation densities for PWIs. Our results ‘partially’ answer one of the fundamental questions posed in [13]—to determine all invariant non-atomic probability Borel measures in piecewise rotations. When restricting PAPs to interval exchange transformations (IETs), our results imply that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have very irregular densities, i.e. they have unbounded variation.

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou–Julia–Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.

We determine all the normal subgroups of the group of Cr diffeomorphisms of ℝn, 1≤r≤∞, except when r=n+1 or n=4, and also of the group of homeomorphisms of ℝn ( r=0). We also study the group A0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with non-empty boundary, then the quotient of A0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.

We construct an example of a Hamiltonian flow $f^t$ on a four-dimensional smooth manifold $\mathcal {M}$ which after being restricted to an energy surface $\mathcal {M}_e$ demonstrates essential coexistence of regular and chaotic dynamics, that is, there is an open and dense $f^t$ -invariant subset $U\subset \mathcal {M}_e$ such that the restriction $f^t|U$ has non-zero Lyapunov exponents in all directions (except for the direction of the flow) and is a Bernoulli flow while, on the boundary $\partial U$ , which has positive volume, all Lyapunov exponents of the system are zero.

We find analytic solutions $\varphi_{t}$ to the conjugation equation $$ f_{t} \circ \varphi_{t}=\varphi_{t} \circ f_{0}, $$ where $f_{t}$ and $\varphi_{t}$ are analytic diffeomorphisms from ${\Bbb T}^{n}$ to itself that depend analytically on $t$, and $f_{0}$ is an Anosov toral diffeomorphism with reduced spectrum and which is diagonalizable.

Stable accessibility of partially hyperbolic systems is central to their stable ergodicity, and we establish its $C^1$ -density among partially hyperbolic flows, as well as in the categories of volume-preserving, symplectic, and contact partially hyperbolic flows. As applications, we obtain on one hand in each of these four categories of flows the $C^1$ -density of the $C^1$ -stable topological transitivity and triviality of the centralizer, and on the other hand the $C^1$ -density of the $C^1$ -stable K-property of the natural volume in the latter three categories.

Consider the skew product $F:\mathbb{T}^{2}\rightarrow \mathbb{T}^{2}$ , $F(x,y)=(f(x),y+\unicode[STIX]{x1D70F}(x))$ , where $f:\mathbb{T}^{1}\rightarrow \mathbb{T}^{1}$ is a piecewise $\mathscr{C}^{1+\unicode[STIX]{x1D6FC}}$ expanding map on a countable partition and $\unicode[STIX]{x1D70F}:\mathbb{T}^{1}\rightarrow \mathbb{R}$ is piecewise $\mathscr{C}^{1}$ . It is shown that if $\unicode[STIX]{x1D70F}$ is not Lipschitz-cohomologous to a piecewise constant function on the joint partition of $f$ and $\unicode[STIX]{x1D70F}$ , then $F$ is mixing at a stretched-exponential rate.