Given a twist map on the torus, we prove that the non-existence of a rotational invariant circle implies that the shear-rotation interval is non-trivial, i.e. the existence of orbits with non-zero mean acceleration. Under the same assumption, we prove the existence of infinitely many periodic orbits and positive entropy invariant measures with non-zero shear-rotation number. The method is variational, but relies on topological understanding of the shear-rotation interval of torus homeomorphisms homotopic to a Dehn twist.

Foreman and Weiss [Measure preserving diffeomorphisms of the torus are unclassifiable. Preprint, 2020, arXiv:1705.04414] obtained an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphism, by using a functor $\mathcal {F}$ (see [Foreman and Weiss, From odometers to circular systems: a global structure theorem. J. Mod. Dyn.15 (2019), 345–423]) mapping odometer-based systems, $\mathcal {OB}$ , to circular systems, $\mathcal {CB}$ . This functor transfers the classification problem from $\mathcal {OB}$ to $\mathcal {CB}$ , and it preserves weakly mixing extensions, compact extensions, factor maps, the rank-one property, and certain types of isomorphisms. Thus it is natural to ask whether $\mathcal {F}$ preserves other dynamical properties. We show that $\mathcal {F}$ does not preserve the loosely Bernoulli property by providing positive and zero-entropy examples of loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli. We also construct a loosely Bernoulli circular system whose corresponding odometer-based system has zero entropy and is not loosely Bernoulli.

We study C1 invariant, two-dimensional tori of constant energy without singularities of the Euler–Lagrange flow of a convex, superlinear Lagrangian defined in the tangent space of a closed surface. We show that in regular energy levels, such tori are graphs of the canonical projection if and only if they are minimizing. This result was known for C3 invariant, Lagrangian tori of Finsler Lagrangians, which implied the statement for C3 invariant Lagrangian tori of unimodal Lagrangians with a high-energy level. We also show that graphs have energy $E \geq c_0(L)$, where c0(L) is Mañé's strict critical value; and that the presence of Reeb components in such tori implies that the energy is c0(L). Moreover, if we assume that the Lagrangian is unimodal, we show that Lagrangian invariant tori with no singularities have no Reeb components, regardless of the energy level.

We use Lerman’s contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed $3$ -manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the $2$ -disc constructed by Fayad and Katok embed into the Reeb flow of a dynamically convex contact form on the $3$ -sphere.

Let $\unicode[STIX]{x1D714}=[a_{1},a_{2},\ldots ]$ be the infinite expansion of a continued fraction for an irrational number $\unicode[STIX]{x1D714}\in (0,1)$, and let $R_{n}(\unicode[STIX]{x1D714})$ (respectively, $R_{n,k}(\unicode[STIX]{x1D714})$, $R_{n,k+}(\unicode[STIX]{x1D714})$) be the number of distinct partial quotients, each of which appears at least once (respectively, exactly $k$ times, at least $k$ times) in the sequence $a_{1},\ldots ,a_{n}$. In this paper, it is proved that, for Lebesgue almost all $\unicode[STIX]{x1D714}\in (0,1)$ and all $k\geq 1$,

$$\begin{eqnarray}\displaystyle \lim _{n\rightarrow \infty }\frac{R_{n}(\unicode[STIX]{x1D714})}{\sqrt{n}}=\sqrt{\frac{\unicode[STIX]{x1D70B}}{\log 2}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n}(\unicode[STIX]{x1D714})}=\frac{C_{2k}^{k}}{(2k-1)\cdot 4^{k}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n,k+}(\unicode[STIX]{x1D714})}=\frac{1}{2k}.\end{eqnarray}$$

$R_{n}$

We construct an infinite family of odd-symplectic forms (also known as Hamiltonian structures) on the $3$-sphere $S^{3}$ that do not admit a symplectic cobordism to the standard contact structure on $S^{3}$. This answers in the negative a question raised by Joel Fish motivated by the search for minimal characteristic flows.

We completely classify the extreme tracial states on the cores of the ${C}^{\ast } $-algebras associated with self-similar maps on compact metric spaces. We present a complete list of them. The extreme tracial states are the union of the discrete type tracial states given by measures supported on the finite orbits of the branch points and a continuous type tracial state given by the Hutchinson measure on the original self-similar set.

Consider $\mathscr{F}$, a non-empty set of subsets of $\mathbb{N}$. An operator $T$ on $X$ satisfies property ${\mathcal{P}}_{\mathscr{F}}$ if, for any non-empty open set $U$ in $X$, there exists $x\in X$ such that $\{n\geq 0:T^{n}x\in U\}\in \mathscr{F}$. Let $\overline{{\mathcal{B}}{\mathcal{D}}}$ be the collection of sets in $\mathbb{N}$ with positive upper Banach density. Our main result is a characterization of a sequence of operators satisfying property ${\mathcal{P}}_{\overline{{\mathcal{B}}{\mathcal{D}}}}$, for which we have used a deep result of Bergelson and McCutcheon in the vein of Szemerédi’s theorem. It turns out that operators having property ${\mathcal{P}}_{\overline{{\mathcal{B}}{\mathcal{D}}}}$ satisfy a kind of recurrence described in terms of essential idempotents of $\unicode[STIX]{x1D6FD}\mathbb{N}$. We will also discuss the case of weighted backward shifts. Finally, we obtain a characterization of reiteratively hypercyclic operators.

A theorem of Łojasiewicz asserts that any relatively compact solution of a real analytic gradient vector field has finite length. We show here a generalization of this result for relatively compact solutions of an analytic vector field X with a smooth invariant hypersurface, transversally hyperbolic for X, where the restriction of the field is a gradient. This solves some instances of R. Thom's Gradient Conjecture. Furthermore, if the dimension of the ambient space is three, these solutions do not oscillate (in the sense that they cut an analytic set only finitely many times); this can also be applied to some gradient vector fields.

A graphic flow is a totally minimal flow such that the only minimal subsets of the product flow are the graphs of the powers of the defining homeomorphism [2]. We consider flows of the form $(X^{k},T^{L})$, where $(X,T)$ is graphic, $k$ is a positive integer, and $L:\{1,\ldots,k\}\to {\Bbb Z}\setminus \{0\}$. It is shown that the isomorphism classes of these flows are determined by the cardinality of $L^{-1}(p)$.

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length $n$ relates to the one for words of length $n+1$ for each $n=1,2,3,\ldots \,$.

Let $f : V \to U$ be a (generalized) polynomial-like map. Suppose that harmonic measure $\omega= \omega(\cdot,\infty)$ on the Julia set $J_{f}$ is equal to the measure of maximal entropy $m$ for $f : J_{f} \hookleftarrow$. Then the dynamics $(f,V,U)$ is called maximal. We are going to give a criterion for the dynamics to be conformally equivalent to a maximal one, that is to be conformally maximal. In the second part of this paper we construct an invariant ‘harmonic’ measure $\mu$ such that ${d\mu}/{d\omega}$ is Hölder for certain dynamics. This allows us to prove in this class of dynamical systems that $\omega\approx m$ is necessary and sufficient for $(f,V,U)$ to be conformally maximal. In the particular case when $f$ is expanding and $J_{f}$ is a circle, our result becomes a theorem of Shub and Sullivan; so throughout the paper we are dealing with an analog of a theorem of Shub and Sullivan on ‘wild’ (e.g. totally disconnected) $J_{f}$ and for certain non-expanding $f$. We also construct (under certain assumptions) invariant harmonic measure on $J_{f}$. In this respect, our work stems from one of the works of Carleson.

We provide a complete characterization of periodic point free homeomorphisms of the $2$ -torus admitting irrational circle rotations as topological factors. Given a homeomorphism of the $2$ -torus without periodic points and exhibiting uniformly bounded rotational deviations with respect to a rational direction, we show that annularity and the geometry of its non-wandering set are the only possible obstructions for the existence of an irrational circle rotation as topological factor. Through a very precise study of the dynamics of the induced $\rho $ -centralized skew-product, we extend and generalize considerably previous results of Jäger.

We prove that if G is a countable discrete group with property (T) over an infinite subgroup H≤G which contains an infinite Abelian subgroup or is normal, then G has continuum-many orbit-inequivalent measure-preserving almost-everywhere-free ergodic actions on a standard Borel probability space. Further, we obtain that the measure-preserving almost-everywhere-free ergodic actions of such a G cannot be classified up to orbit equivalence by a reasonable assignment of countable structures as complete invariants. We also obtain a strengthening and a new proof of a non-classification result of Foreman and Weiss for conjugacy of measure-preserving ergodic almost-everywhere-free actions of discrete countable groups.

The critical dimension of an ergodic non-singular dynamical system is the asymptotic growth rate of sums of consecutive Radon–Nikodým derivatives. This has been shown to equal the average coordinate entropy for product odometers when the size of individual factors is bounded. We extend this result to $G$-measures with an asymptotic bound on the size of individual factors. Furthermore, unlike von Neumann–Krieger type, the critical dimension is an invariant property on the class of ergodic $G$-measures.

Given an n-dimensional substitution Φ whose associated linear expansion Λ is unimodular and hyperbolic, we use elements of the one-dimensional integer Čech cohomology of the tiling space ΩΦ to construct a finite-to-one semi-conjugacy G:ΩΦ→𝕋D, called a geometric realization, between the substitution induced dynamics and an invariant set of a hyperbolic toral automorphism. If Λ satisfies a Pisot family condition and the rank of the module of generalized return vectors equals the generalized degree of Λ, G is surjective and coincides with the map onto the maximal equicontinuous factor of the ℝn-action on ΩΦ. We are led to formulate a higher-dimensional generalization of the Pisot substitution conjecture: if Λ satisfies the Pisot family condition and the rank of the one-dimensional cohomology of ΩΦ equals the generalized degree of Λ, then the ℝn-action on ΩΦhas pure discrete spectrum.

Let r, s be natural numbers, r, s \geq 2, such that {\rm log}\,r/{\rm log}\,s is irrational. Let \mu_j = \sum_{k=0}^{s-1} p^{(j)}_{k}\delta_{k/s^j}, where p^{(j)}_k \in [0,1), \sum_{k=0}^{s-1} p^{(j)}_k =1, j = 1, 2, \dotsc and {\delta}_{x} denotes the probability atom at x. We prove that if there exists k_0\in \{0,\dotsc , s-2\}, such that \inf \{ p^{(j)}_{k_0}, p^{(j)}_{k_0+1} : j\in \mathbb N \} > 0 and \mu is the infinite convolution measure \mu = *^{\infty}_{j=1} {\mu_{j} }, where the convergence is in the weak* sense, then \mu-almost all numbers are normal to base r. This generalizes some earlier results of Feldman, Smorodinsky and Pushkin.

Let $\phi :X\to X$ be a homeomorphism of a compact metric space X. For any continuous function $F:X\to \mathbb {R}$ there is a one-parameter group $\alpha ^{F}$ of automorphisms (or a flow) on the crossed product $C^*$ -algebra $C(X)\rtimes _{\phi }\mathbb {Z}$ defined such that $\alpha ^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and U is the canonical unitary in the construction of the crossed product. In this paper we study the Kubo--Martin--Schwinger (KMS) states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when $C(X) \rtimes _{\phi } \mathbb Z$ is simple this set is either $\{0\}$ or the whole line $\mathbb R$ .

We study limit sets of stable cellular automata from a symbolic dynamics point of view, where they are a special case of sofic shifts admitting a steady epimorphism. We prove that there exists a right-closing almost-everywhere steady factor map from one irreducible sofic shift onto another one if and only if there exists such a map from the domain onto the minimal right-resolving cover of the image. We define right-continuing almost-everywhere steady maps, and prove that there exists such a steady map between two sofic shifts if and only if there exists a factor map from the domain onto the minimal right-resolving cover of the image. To translate this into terms of cellular automata, a sofic shift can be the limit set of a stable cellular automaton with a right-closing almost-everywhere dynamics onto its limit set if and only if it is the factor of a full shift and there exists a right-closing almost-everywhere factor map from the sofic shift onto its minimal right-resolving cover. A sofic shift can be the limit set of a stable cellular automaton reaching its limit set with a right-continuing almost-everywhere factor map if and only if it is the factor of a full shift and there exists a factor map from the sofic shift onto its minimal right-resolving cover. Finally, as a consequence of the previous results, we provide a characterization of the almost of finite type shifts (AFT) in terms of a property of steady maps that have them as range.

We consider singular perturbation of a mixing subshift of finite type by means of thermodynamic formalism. In our formulation, the perturbed systems are described by a family of potentials {Φ(α,⋅)} with large parameter α on a fixed subshift of finite type, and the original (unperturbed) system is characterized as the system at infinity obtained by collapsing the perturbed system upon taking . We apply our formulation to the collapse of cookie-cutter systems and dispersing open billiards.