We confirm a conjecture posed by Bergelson, Moreira, and Richter, and in particular show that for every probability-measure-preserving system $(X,\mathscr {B},\mu ,T)$, every $k\in \mathbb {N}$, every set $A\in \mathscr {B}$ with $\mu (A)>0$, and every tempered function f, $ \lim _{N\to \infty }({1}/{N}) \sum _{n=1}^N\mu (A\cap T^{-\lfloor {f(n)\rfloor }}A\cap T^{-\lfloor {f(n+1)\rfloor }}A\cap \cdots \cap T^{-\lfloor {f(n+k)\rfloor }}A)>0. $ This is achieved by establishing conditions on an increasing function $W:\mathbb {N}\rightarrow (0,\infty )$ such that if $(x_n)_{n\in \mathbb {N}}$ is a bounded sequence in a Banach space with $ \lim _{W(N)-W(M)\to \infty }({1}/({W(N)-W(M)})) \sum _{n=M}^N (W(n)-W(n-1))x_n =L $ then the limit of Cesàro averages of $(x_n)_{n\in \mathbb {N}}$, $\lim _{N\to \infty }({1}/{N})\sum _{n=1}^Nx_n$, is also equal to L. Furthermore, the methods we develop can be used to sharpen some of the combinatorial results obtained by Bergelson, Moreira, and Richter. For example, if E is a set of positive upper density, then for any $k\in \mathbb {N}$, any $\varepsilon>0$, and all sufficiently large $N\in \mathbb {N}$ there is an $a\in \mathbb {N}$ and an $n\in [N-N^{1/2+\varepsilon },N]$ such that $\{a,a+\lfloor n^{3/2} \rfloor ,a+\lfloor (n+1)^{3/2}\rfloor ,\ldots ,a +\lfloor (n+k)^{3/2}\rfloor \}\subseteq E.$

We find necessary and sufficient conditions for high-order persistence of resonant caustics in perturbed circular billiards. The main tool is a perturbation theory based on the Bialy–Mironov generating function for convex billiards. All resonant caustics with period q persist up to order $\lceil q/n \rceil -1$ under any polynomial deformation of the circle of degree n.

We prove that, for a $C^2$ partially hyperbolic endomorphism of the 2-torus which is strongly transitive, given an ergodic u-Gibbs measure that has positive center Lyapunov exponent and has full support, then either the map is special (has only one unstable direction per point) or the measure is the unique absolutely continuous invariant measure. We can apply this result in many settings, in particular, obtaining uniqueness of u-Gibbs measures for every non-special perturbation of irreducible linear expanding maps of the torus with simple spectrum. This gives new open sets of partially hyperbolic systems displaying a unique u-Gibbs measure.

We prove that in the space of $C^r$ maps $(r=2,\ldots ,\infty ,\omega )$ of a smooth manifold of dimension at least 4, there exist open regions where maps with infinitely many corank-2 homoclinic tangencies of all orders are dense. The result is applied to show the existence of maps with universal two-dimensional dynamics, that is, maps whose iterations approximate the dynamics of every map of a two-dimensional disk with an arbitrarily good accuracy. We show that maps with universal two-dimensional dynamics are $C^r$-generic in the regions under consideration.

In this paper, we study the quantitative recurrence properties in the case of $\mathbb {Z}$-extension of Axiom A flows on a Riemannian manifold. We study the asymptotic behavior of the first return time to a small neighborhood of the starting point. We establish results of almost everywhere convergence, and of convergence in distribution with respect to any probability measure absolutely continuous with respect to the infinite invariant measure. In particular, our results apply to geodesic flows on the $\mathbb {Z}$-cover of compact smooth surfaces of negative curvature.

We establish a polynomial ergodic theorem for actions of the affine group of a countable field K. As an application, we deduce—via a variant of Furstenberg’s correspondence principle—that for fields of characteristic zero, any ‘large’ set $E\subset K$ contains ‘many’ patterns of the form $\{p(u)+v,uv\}$, for every non-constant polynomial $p(x)\in K[x]$. Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a finitistic variant of Bergelson’s ‘colouring trick’, show that for $r\in \mathbb N$ fixed, any r-colouring of a large enough finite field will contain monochromatic patterns of the form $\{u,p(u)+v,uv\}$. In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalization of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic ‘colouring trick’, we provide a conditional, elementary generalization of Green and Sanders’ $\{u,v,u+v,uv\}$ theorem.

Let $X=G/\Gamma $ be the quotient of a semisimple Lie group G by its non-cocompact arithmetic lattice. Let H be a reductive algebraic subgroup of G acting on X. We give several equivalent algebraic conditions on H for the existence of a fixed compact set in X intersecting every H-orbit. This generalizes previous results concerning certain special reductive group action on X in this setting. When G is of real rank one, $\Gamma $ is a non-cocompact lattice of G, and $H<G$ is an algebraic group, we also obtain an algebraic condition on H which is equivalent to the return of every H-orbit to a single compact set in X. This complements our results in the case of an arithmetic lattice.

In this article, we consider a closed rank-one Riemannian manifold M without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on M with length at most t, and $\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates:

$$ \begin{align*}\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\end{align*} $$

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of continuous functions on the shift space contains two disjoint subsets: one is a dense $G_\delta $ set for which all maximizing measures have ‘relatively small’ entropy; the other is the set of functions having uncountably many, fully supported ergodic maximizing measures with ‘relatively large’ entropy. This result generalizes and unifies the results of Morris [Discrete Contin. Dyn. Syst. 27 (2010), 383–388] and Shinoda [Nonlinearity 31 (2018), 2192–2200] on symbolic dynamics, and applies to a wide class of intrinsically ergodic non-Markov symbolic dynamics without the Bowen specification property, including any transitive piecewise monotonic interval map, some coded shifts, and multidimensional $\beta $-transformations. Along with these examples of application, we provide an example of an intrinsically ergodic subshift with positive obstruction entropy to specification.

We consider $L^q$-spectra of planar graph-directed self-affine measures generated by diagonal or anti-diagonal matrices. Assuming the directed graph is strongly connected and the system satisfies the rectangular open set condition, we obtain a general closed form expression for the $L^q$-spectra. Consequently, we obtain a closed form expression for box dimensions of associated planar graph-directed box-like self-affine sets. We also provide a precise answer to a question posed by Fraser [On the $L^q$-spectrum of planar self-affine measures. Trans. Amer. Math. Soc. 368(8) (2016), 5579–5620] concerning the $L^q$-spectra of planar self-affine measures generated by diagonal matrices. An interesting observation of the closed form expression is that it is possible to calculate the $L^q$-spectrum of a measure without involving the exact $L^q$-spectra of its projections to the axes.

In this article, we study the pressure at infinity of potentials defined over countable Markov shifts. We establish an upper semi-continuity result concerning the limiting behaviour of the pressure of invariant probability measures, where the escape of mass is controlled by the pressure at infinity. As a consequence, we establish criteria for the existence of equilibrium states and maximizing measures for uniformly continuous potentials. Additionally, we study the pressure at infinity of suspension flows defined over countable Markov shifts and prove an upper semi-continuity result for the pressure map.

We establish a relation between the continuity of the fiber entropy and the continuity of the fiber Lyapunov exponents for skew products with two-dimensional fibers. This result extends the theorem for surfaces proved by Buzzi, Crovisier, and Sarig. As a consequence, we are able to obtain classes of skew products that satisfy the strong positive recurrence property, in particular, these maps have finite number of measures of maximal entropy, all exponentially mixing with good statistical properties.

We study the probability that an AR(1) Markov chain $X_{n+1}=aX_n+\xi _{n+1}$, where $a\in (0,1)$ is a constant, stays non-negative for a long time. We find the exact asymptotics of this probability and the weak limit of $X_n$ conditioned to stay non-negative, assuming that the independent and identically distributed innovations $\xi _n$ take only two values $\pm 1$ and $a \le \tfrac 23$. This limiting distribution is quasi-stationary. It has no atoms and is singular with respect to the Lebesgue measure when $\tfrac 12< a \le \tfrac 23$, except for the case when $a=\tfrac 23$ and $\mathbb P(\xi _n=1)=\tfrac 12$, where this distribution is uniform on the interval $[0,3]$. This is similar to the properties of Bernoulli convolutions. For $0 < a \le \tfrac 12$, the situation is much simpler and the limiting distribution is a $\delta $-measure. To prove these results, we uncover a close connection between $X_n$ killed at exiting $[0, \infty )$ and the classical dynamical system defined by the piecewise linear mapping $x \mapsto x/a + 1/2\ \pmod 1$. Namely, the trajectory of this system started at $X_n$ deterministically recovers the values of the killed chain in reversed time. We use this fact to construct a suitable Banach space, where the transition operator of the killed chain has the compactness properties that allow us to apply a conventional argument of the Perron–Frobenius type.

The almost sure convergence of ergodic averages in Birkhoff’s pointwise ergodic theorem is known to fail in the finitely additive setting. We introduce a natural reformulation of almost sure convergence suitable for finitely additive measures, which we call finite almost sure convergence. Unlike the classical formulation, finite almost sure convergence only involves measures of finite unions and intersections, making it well adapted to finitely additive spaces. Using this notion, we extend the pointwise ergodic theorem to finitely additive probability spaces. Our proof relies on demonstrating that several quantitative generalizations of the pointwise ergodic theorem remain valid in the finitely additive setting via an extension of the Calderón transference principle. The result then follows by exploiting the relationships between quantitative notions of almost sure convergence developed by the author and Powell [On quantitative convergence for stochastic processes: crossings, fluctuations and martingales. Trans. Amer. Math. Soc. Series B 12 (2025), 974–1019].

A Brouwer homeomorphism is a fixed-point free, orientation-preserving homeomorphism of the plane. A foundational result of Le Calvez [A foliated version of the Brouwer translation theorem. Comment. Math. Helv. 79(2) (2004), 229–259 (in French)] establishes that every such homeomorphism f admits an oriented planar foliation ${\mathcal {F}}$ such that every point $x \in \mathbb R^2$ can be connected to its image $f(x)$ by a path positively transverse to ${\mathcal {F}}$. This provides a powerful framework for analyzing the dynamics of f by studying how its orbits cross the leaves of $\mathcal {F}$. In this article, we refine this framework by identifying additional qualitative dynamical information about f that is encoded in ${\mathcal {F}}$, which can be systematically recovered through the concept of proper transverse trajectories. Later, we investigate the possible combinatorial configurations of these proper trajectories for finite collections of orbits and characterize their simplest forms. As a key application, this refined framework is used in a subsequent work [A foliated viewpoint on homotopy Brouwer theory. Preprint, 2025, arXiv:2512.07510] to offer a new perspective on homotopy Brouwer theory, originally introduced by Handel [A fixed-point theorem for planar homeomorphisms. Topology 38(2) (1999), 235–264].

In 2002, Kamae and Zamboni [Ergod. Th. & Dynam. Sys. 22(4) (2002), 1191–1199] introduced maximal pattern complexity and determined that any aperiodic sequence must have maximal pattern complexity at least $2k$. In 2006, Kamae and Rao [European J. Combin. 27(1) (2006), 125–137] examined the maximal pattern complexity of sequences over larger alphabets and showed that sequences which have maximal pattern complexity less than $\ell k$, for $\ell $ the size of the alphabet, must have some periodic structure. In this paper, we investigate the structure of sequences of low maximal pattern complexity over $\ell $ letters, where $\liminf _{k \to \infty } p_{\alpha }^*(k) - 3k = -\infty $. In addition, we show that the minimal maximal pattern complexity of an aperiodic sequence which uses all $\ell $ letters is $p_{\alpha }^*(k) = 2k + \ell -2$ and give an exact structure for aperiodic sequences with this maximal pattern complexity.