The local structure of rotationally symmetric Finsler surfaces with vanishing flag curvature is completely determined in this paper. A geometric method for constructing such surfaces is introduced. The construction begins with a planar vector field X that depends on two functions of one variable. It is shown that the flow of X could be used to generate a generalized Finsler surface with zero flag curvature. Moreover, this generalized structure reduces to a regular Finsler metric if and only if X has an isochronous center. By relating X to a Liénard system, we obtain the isochronicity condition and discover numerous new examples of complete flat Finsler surfaces, depending on an odd function and an even function.

We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

Based on previous work of the authors, to any S-adic development of a subshift X a ‘directive sequence’ of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given S-adic development. The issuing rich picture enables one to deduce results about X with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result, we also exhibit, for any integer $d \geq 2$, an S-adic development of a minimal, aperiodic, uniquely ergodic subshift X, where all level alphabets $\mathcal A_n$ have cardinality $d,$ while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subseteq \mathcal A_n^{\mathbb {Z}}$.

In this paper, we study the Dirichlet problem for systems of mean value equations on a regular tree. We deal both with the directed case (the equations verified by the components of the system at a node in the tree only involve values of the unknowns at the successors of the node in the tree) and the undirected case (now the equations also involve the predecessor in the tree). We find necessary and sufficient conditions on the coefficients in order to have existence and uniqueness of solutions for continuous boundary data. In a particular case, we also include an interpretation of such solutions as a limit of value functions of suitable two-players zero-sum games.

We study conjugacy classes of germs of nonflat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to the identity. The core of all of this lies in the invariance of residues under low-regular conjugacies. This may be seen as an extension of the fact (also proved in this article) that the value of the Schwarzian derivative at the origin for germs of $C^3$ parabolic diffeomorphisms is invariant under $C^2$ parabolic conjugacy, though it may vary arbitrarily under parabolic $C^1$ conjugacy.

In this paper, we study centrally symmetric Birkhoff billiard tables. We introduce a closed invariant set $\mathcal {M}_{\mathcal {B}}$ consisting of locally maximizing orbits of the billiard map lying inside the region $\mathcal {B}$ bounded by two invariant curves of $4$-periodic orbits. We give an effective bound from above on the measure of this invariant set in terms of the isoperimetric defect of the curve. The equality case occurs if and only if the curve is a circle.

We consider the attractor $\Lambda $ of a piecewise contracting map f defined on a compact interval. If f is injective, we show that it is possible to estimate the topological entropy of f (according to Bowen’s formula) and the Hausdorff dimension of $\Lambda $ via the complexity associated with the orbits of the system. Specifically, we prove that both numbers are zero.

This paper mainly concerns the KAM persistence of the mapping $\mathscr {F}:\mathbb {T}^{n}\times E\rightarrow \mathbb {T}^{n}\times \mathbb {R}^{n}$ with intersection property, where $E\subset \mathbb {R}^{n}$ is a connected closed bounded domain with interior points. By assuming that the frequency mapping satisfies certain topological degree condition and weak convexity condition, we prove some Moser-type results about the invariant torus of mapping $\mathscr {F}$ with frequency-preserving under small perturbations. To our knowledge, this is the first approach to Moser's theorem with frequency-preserving. Moreover, given perturbed mappings over $\mathbb {T}^n$, it is shown that such persistence still holds when the frequency mapping and perturbations are only continuous about parameter beyond Lipschitz or even Hölder type. We also touch the parameter without dimension limitation problem under such settings.

For integers a and $b\geq 2$, let $T_a$ and $T_b$ be multiplication by a and b on $\mathbb {T}=\mathbb {R}/\mathbb {Z}$. The action on $\mathbb {T}$ by $T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if a and b are multiplicatively independent, then the only $\times a,\times b$ invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, it is not known whether there exists a non-trivial $\times a,\times b$ invariant and ergodic measure. In this paper, we study the empirical measures of $x\in \mathbb {T}$ with respect to the $\times a,\times b$ action and show that the set of x such that the empirical measures of x do not converge to any measure has Hausdorff dimension one and the set of x such that the empirical measures can approach a non-trivial $\times a,\times b$ invariant measure has Hausdorff dimension zero. Furthermore, we obtain some equidistribution result about the $\times a,\times b$ orbit of x in the complement of a set of Hausdorff dimension zero.

We study stationary measures for iterated function systems (considered as random dynamical systems) consisting of two piecewise affine interval homeomorphisms, called Alsedà–Misiurewicz (AM) systems. We prove that for an open set of parameters, the unique non-atomic stationary measure for an AM system has Hausdorff dimension strictly smaller than $1$. In particular, we obtain singularity of these measures, answering partially a question of Alsedà and Misiurewicz [Random interval homeomorphisms. Publ. Mat. 58(suppl.) (2014), 15–36].

We study infinite systems of mean field weakly coupled intermittent maps in the Pomeau–Manneville scenario. We prove that the coupled system admits a unique ‘physical’ stationary state, to which all absolutely continuous states converge. Moreover, we show that suitably regular states converge polynomially.

Inspired by a twist map theorem of Mather. we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map g to the k-fold cover. For each irrational in the rotation set’s interior, the collection of the k-fold ordered semi-Denjoy minimal sets with that rotation number contains a $(k-1)$-dimensional ball with the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool used is a generalization of a construction of Hedlund and Morse that generates symbolic analogs of these k-fold well-ordered invariant sets.

A left orderable monster is a finitely generated left orderable group all of whose fixed point-free actions on the line are proximal: the action is semiconjugate to a minimal action so that for every bounded interval I and open interval J, there is a group element that sends I into J. In his 2018 ICM address, Navas asked about the existence of left orderable monsters. By now there are several examples, all of which are finitely generated but not finitely presentable. We provide the first examples of left orderable monsters that are finitely presentable, and even of type $F_\infty $. These groups satisfy several additional properties separating them from the previous examples: they are not simple, they act minimally on the circle, and they have an infinite-dimensional space of homogeneous quasimorphisms. Our construction is flexible enough that it produces infinitely many isomorphism classes of finitely presented (and type $F_{\infty }$) left orderable monsters.

In this paper we study persistence features of topological entropy and periodic orbit growth of Hamiltonian diffeomorphisms on surfaces with respect to Hofer's metric. We exhibit stability of these dynamical quantities in a rather strong sense for a specific family of maps studied by Polterovich and Shelukhin. A crucial ingredient comes from enhancement of lower bounds for the topological entropy and orbit growth forced by a periodic point, formulated in terms of the geometric self-intersection number and a variant of Turaev's cobracket of the free homotopy class that it induces. Those bounds are obtained within the framework of Le Calvez and Tal's forcing theory.

The action on the trace space induced by a generic automorphism of a suitable finite classifiable ${\mathrm {C}^*}$ -algebra is shown to be chaotic and weakly mixing. Model ${\mathrm {C}^*}$ -algebras are constructed to observe the central limit theorem and other statistical features of strongly chaotic tracial actions. Genericity of finite Rokhlin dimension is used to describe $KK$ -contractible stably projectionless ${\mathrm {C}^*}$ -algebras as crossed products.

As a weak version of embedding flow, the problem of iterative roots is studied extensively in one dimension, especially in monotone case. There are few results in high dimensions because the constructive method dealing with monotone mappings is unavailable. In this paper, by introducing a kind of partial order, we define the monotonicity for two-dimensional mappings and then present some results on the existence of iterative roots for linear mappings, triangle-type mappings, and co-triangle-type mappings, respectively. Our theorems show that even the property of monotonicity for iterative roots of monotone mappings, which is a trivial result in one dimension, does not hold anymore in high dimensions. At the end of this paper, the problem of iterative roots for two well-known planar mappings, that is, Hénon mappings and coupled logistic mappings, are also discussed.

We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of hameomorphisms constructed by Oh and Müller, and we construct an infinite-dimensional family of quasi-morphisms on the group of area and orientation preserving homeomorphisms of the two-sphere.

Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on ‘classical’ Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms).

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.

Open conjectures state that, for every $x\in [0,1]$, the orbit $(x_n)_{n=1}^\infty $ of the mean-median recursion

$$ \begin{align*}x_{n+1}=(n+1)\cdot\operatorname{\mathrm{median}}(x_1,\ldots,x_{n})-(x_1+\cdots+x_n),\quad n\geqslant 3,\end{align*} $$

with initial data $(x_1,x_2,x_3)=(0,x,1)$, is eventually constant, and that its transit time and limit functions (of x) are unbounded and continuous, respectively. In this paper, we prove that for the slightly modified recursion

$$ \begin{align*}x_{n+1}=n\cdot\operatorname{\mathrm{median}}(x_1,\ldots,x_{n})-(x_1+\cdots+x_n),\quad n\geqslant 3,\end{align*} $$

first suggested by Akiyama, the transit time function is unbounded but the limit function is discontinuous.

We prove an explicit characterization of the points in Thurston’s Master Teapot, which can be implemented algorithmically to test whether a point in $\mathbb {C}\times \mathbb {R}$ belongs to the complement of the Master Teapot. As an application, we show that the intersection of the Master Teapot with the unit cylinder is not symmetrical under reflection through the plane that is the product of the imaginary axis of $\mathbb {C}$ and $\mathbb {R}$.