In his last paper, William Thurston defined the Master Teapot as the closure of the set of pairs $(z,s)$, where s is the slope of a tent map $T_s$ with the turning point periodic, and the complex number z is a Galois conjugate of s. In this case $1/z$ is a zero of the kneading determinant of $T_s$. We remove the restriction that the turning point is periodic, and sometimes look beyond tent maps. However, we restrict our attention to zeros $x=1/z$ in the real interval $(0,1)$. By the results of Milnor and Thurston, the kneading determinant has such a zero if and only if the map has positive topological entropy. We show that the first (smallest) zero is simple, but among other zeros there may be multiple ones. We describe a class of unimodal maps, so-called R-even ones, whose kneading determinant has only one zero in $(0,1)$. In contrast, we show that generic mixing tent maps have kneading determinants with infinitely many zeros in $(0,1)$. We prove that the second zero in $(0,1)$ of the kneading determinant of a unimodal map, provided it exists, is always greater than or equal to $\sqrt [3]{1/2}$, and if the kneading sequence begins with $RL^NR$, $N\geq 2$, then the best lower bound for the second zero is in fact $\sqrt [N+1]{1/2}$. We also investigate (partially numerically) the shape of the Real Teapot, consisting of the pairs $(s,x)$, where x in $(0,1)$ is a zero of the kneading determinant of $T_s$, and $s\in (1,2]$.

An area-preserving homeomorphism isotopic to the identity is said to have rational rotation direction if its rotation vector is a real multiple of a rational class. We give a short proof that any area-preserving homeomorphism of a compact surface of genus at least two, which is isotopic to the identity and has rational rotation direction, is either the identity or has periodic points of unbounded minimal period. This answers a question of Ginzburg and Seyfaddini and can be regarded as a Conley conjecture-type result for symplectic homeomorphisms of surfaces beyond the Hamiltonian case. We also discuss several variations, such as maps preserving arbitrary Borel probability measures with full support, maps that are not isotopic to the identity and maps on lower genus surfaces. The proofs of the main results combine topological arguments with periodic Floer homology.

We study the problem of conjugating a diffeomorphism of the interval to (positive) powers of itself. Although this is always possible for homeomorphisms, the smooth setting is rather interesting. Besides the obvious obstruction given by hyperbolic fixed points, several other aspects need to be considered. As concrete results we show that, in class C1, if we restrict to the (closed) subset of diffeomorphisms having only parabolic fixed points, the set of diffeomorphisms that are conjugate to their powers is dense, but its complement is generic. In higher regularity, however, the complementary set contains an open and dense set. The text is complemented with several remarks and results concerning distortion elements of the group of diffeomorphisms of the interval in several regularities.

Consider, for any integer $n\ge 3$, the set $\operatorname {\mathrm {Pos}}_n$ of all n-periodic tree patterns with positive topological entropy and the set $\operatorname {\mathrm {Irr}}_n\subset \operatorname {\mathrm {Pos}}_n$ of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\operatorname {\mathrm {Pos}}_n$, $\operatorname {\mathrm {Irr}}_n$ and $\operatorname {\mathrm {Pos}}_n\setminus \operatorname {\mathrm {Irr}}_n$. Let $\unicode{x3bb} _n$ be the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty )$. We explicitly construct an irreducible n-periodic tree pattern $\mathcal {Q}_n$ whose entropy is $\log (\unicode{x3bb} _n)$. We prove that this entropy is minimum in $\operatorname {\mathrm {Pos}}_n$. Since the pattern $\mathcal {Q}_n$ is irreducible, $\mathcal {Q}_n$ also minimizes the entropy in the family $\operatorname {\mathrm {Irr}}_n$. We also prove that the minimum positive entropy in the set $\operatorname {\mathrm {Pos}}_n\setminus \operatorname {\mathrm {Irr}}_n$ (which is non-empty only for composite integers $n\ge 6$) is $\log (\unicode{x3bb} _{n/p})/p$, where p is the least prime factor of n.

We study the global behavior of the renormalization operator on a specially constructed Banach manifold that has cubic critical circle maps on its boundary and circle diffeomorphisms in its interior. As an application, we prove results on smoothness of irrational Arnold tongues.

We prove that almost every interval exchange transformation, with an associated translation surface of genus $g\geq 2$, can be non-trivially and isometrically embedded in a family of piecewise isometries. In particular, this proves the existence of invariant curves for piecewise isometries, reminiscent of Kolmogorov–Arnold–Moser (KAM) curves for area-preserving maps, which are not unions of circle arcs or line segments.

For a class of potentials $\psi $ satisfying a condition depending on the roof function of a suspension (semi)flow, we show an EKP inequality, which can be interpreted as a Hölder continuity property in the weak${^*}$ norm of measures, with respect to the pressure of those measures, where the Hölder exponent depends on the $L^q$-space to which $\psi $ belongs. This also captures a new type of phase transition for intermittent (semi)flows (and maps).

Quasigeodesic behavior of flow lines is a very useful property in the study of Anosov flows. Not every Anosov flow in dimension three is quasigeodesic. In fact, until recently, up to orbit equivalence, the only previously known examples of quasigeodesic Anosov flows were suspension flows. In a recent article, the second author proved that an Anosov flow on a hyperbolic 3-manifold is quasigeodesic if and only if it is non-$\mathbb {R}$-covered, and this result completes the classification of quasigeodesic Anosov flows on hyperbolic 3-manifolds. In this article, we prove that a new class of examples of Anosov flows are quasigeodesic. These are the first examples of quasigeodesic Anosov flows on 3-manifolds that are neither Seifert, nor solvable, nor hyperbolic. In general, it is very hard to show that a given flow is quasigeodesic and, in this article, we provide a new method to prove that an Anosov flow is quasigeodesic.

Consider a flow in $\mathbb{R}^3$ and let K be the biggest invariant subset of some compact region of interest $N \subseteq \mathbb{R}^3$. The set K is often not computable, but the way the flow crosses the boundary of N can provide indirect information about it. For example, classical tools such as Ważewski’s principle or the Poincaré–Hopf theorem can be used to detect whether K is non-empty or contains rest points, respectively. We present a criterion that can establish whether K has a non-trivial homology by looking at the subset of the boundary of N along which the flow is tangent to N. We prove that the criterion is as sharp as possible with the information it uses as an input. We also show that it is algorithmically checkable.

We study a skew product transformation associated to an irrational rotation of the circle $[0,1]/\sim $. This skew product keeps track of the number of times an orbit of the rotation lands in the two complementary intervals of $\{0,1/2\}$ in the circle. We show that under certain conditions on the continued fraction expansion of the irrational number defining the rotation, the skew product transformation has certain dense orbits. This is in spite of the presence of numerous non-dense orbits. We use this to construct laminations on infinite type surfaces with exotic properties. In particular, we show that for every infinite type surface with an isolated planar end, there is an infinite clique of $2$-filling rays based at that end. These $2$-filling rays are relevant to Bavard and Walker’s loop graphs.

Let $f,g$ be $C^2$ expanding maps on the circle which are topologically conjugate. We assume that the derivatives of f and g at corresponding periodic points coincide for some large period N. We show that f and g are ‘approximately smoothly conjugate.’ Namely, we construct a $C^2$ conjugacy $h_N$ such that $h_N$ is exponentially close to h in the $C^0$ topology, and $f_N:=h_N^{-1}gh_N$ is exponentially close to f in the $C^1$ topology. Our main tool is a uniform effective version of Bowen’s equidistribution of weighted periodic orbits to the equilibrium state.

Let $M$ be an oriented smooth manifold and $\operatorname{Homeo}\!(M,\omega )$ the group of measure preserving homeomorphisms of $M$, where $\omega$ is a finite measure induced by a volume form. In this paper, we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0\!(M,\omega )$ and $\operatorname{Homeo}_+\!(M,\omega )$, respectively, and in several cases prove their non-triviality. More precisely, we define:

• • Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0\!(M,\omega ))$, where $M$ is a hyperbolic manifold of dimension $n$.

• • Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}_+(S,\omega ))$, where $S$ is an oriented closed hyperbolic surface.

We show that Euler classes have positive norms for any closed hyperbolic surface and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$-manifolds; hence, they are non-trivial.

We study the dynamics of dissipative billiard maps within planar convex domains. Such maps have a global attractor. We are interested in the topological and dynamical complexity of the attractor, in terms both of the geometry of the billiard table and of the strength of the dissipation. We focus on the study of an invariant subset of the attractor, the so-called Birkhoff attractor. On the one hand, we show that for a generic convex table with ‘pinched’ curvature, the Birkhoff attractor is a normally contracted manifold when the dissipation is strong. On the other hand, for a mild dissipation, we prove that, generically, the Birkhoff attractor is complicated, both from the topological and the dynamical points of view.

Consider the quadratic family $T_a(x) = a x (1 - x)$ for $x \in [0, 1]$ and mixing Collet–Eckmann (CE) parameters $a \in (2,4)$. For bounded $\varphi $, set $\tilde \varphi _{a} := \varphi - \int \varphi \, d\mu _a$, with $\mu _a$ the unique acim of $T_a$, and put $(\sigma _a (\varphi ))^2 := \int \tilde \varphi _{a}^2 \, d\mu _a + 2 \sum _{i>0} \int \tilde \varphi _{a} (\tilde \varphi _{a} \circ T^i_{a}) \, d\mu _a$. For any mixing Misiurewicz parameter $a_{*}$, we find a positive measure set $\Omega _{*}$ of mixing CE parameters, containing $a_{*}$ as a Lebesgue density point, such that for any Hölder $\varphi $ with $\sigma _{a_{*}}(\varphi )\ne 0$, there exists $\epsilon _\varphi>0$ such that, for normalized Lebesgue measure on $\Omega _{*}\cap [a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$, the functions $\xi _i(a)=\tilde \varphi _a(T_a^{i+1}(1/2))/\sigma _a (\varphi )$ satisfy an almost sure invariance principle (ASIP) for any error exponent $\gamma>2/5$. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann’s proof for piecewise expanding maps. We need to introduce a variant of Benedicks–Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from Baladi et al [Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201 (2015), 773–844].

Let G be a torsion-free, finitely generated, nilpotent and metabelian group. In this work, we show that G embeds into the group of orientation-preserving $C^{1+\alpha }$-diffeomorphisms of the compact interval for all $\alpha < 1/k$, where k is the torsion-free rank of $G/A$ and A is a maximal abelian subgroup. We show that, in many situations, the corresponding $1/k$ is critical in the sense that there is no embedding of G with higher regularity. A particularly nice family where this happens is the family of $(2n+1)$-dimensional Heisenberg groups, for which we can show that the critical regularity is equal to $1+1/n$.

We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized Poincaré exponent of the Fuchsian group. We employ symbolic dynamics developed by Bowen and Series, ergodic theory and thermodynamic formalism to prove the analyticity of the dimension spectrum.

Let $h : \mathbb{R}^2 \to \mathbb{R}^2$ be an orientation preserving homeomorphism of the plane. For any bounded orbit $\mathcal{O}(x)=\{h^n(x):n\in\mathbb{Z}\}$ there exists a fixed point $p\in\mathbb{R}^2$ of h linked to $\mathcal{O}(x)$ in the sense of Gambaudo: one cannot find a Jordan curve $C\subseteq\mathbb{R}^2$ around $\mathcal{O}(x)$, separating it from p, that is isotopic to h(C) in $\mathbb{R}^2\setminus\left(\mathcal{O}(x)\cup\{p\}\right)$.

We prove the Girth Alternative for finitely generated subgroups of $PL_o(I)$. We also prove that a finitely generated subgroup of Homeo$_{+}(I)$ which is sufficiently rich with hyperbolic-like elements has infinite girth.

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.