In her thesis, Mirzakhani showed that the number of simple closed geodesics of length $\leq L$ on a closed, connected, oriented hyperbolic surface X of genus g is asymptotic to $L^{6g-6}$ times a constant depending on the geometry of X. In this survey, we give a detailed account of Mirzakhani’s proof of this result aimed at non-experts. We draw inspiration from classic primitive lattice point counting results in homogeneous dynamics. The focus is on understanding how the general principles that drive the proof in the case of lattices also apply in the setting of hyperbolic surfaces.

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.

We develop combinatorial tools to study partial rigidity within the class of minimal $\mathcal {S}$-adic subshifts. By leveraging the combinatorial data of well-chosen Kakutani–Rokhlin partitions, we establish a necessary and sufficient condition for partial rigidity. Additionally, we provide an explicit expression to compute the partial rigidity rate and an associated partial rigidity sequence. As applications, we compute the partial rigidity rate for a variety of constant length substitution subshifts, such as the Thue–Morse subshift, where we determine a partial rigidity rate of 2/3. We also exhibit non-rigid substitution subshifts with partial rigidity rates arbitrarily close to 1 and, as a consequence, using products of the aforementioned substitutions, we obtain that any number in $[0, 1]$ is the partial rigidity rate of a system.

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.

Many continua that admit a transitive homeomorphism may be found in the literature. The circle is probably the simplest non-degenerate continuum that admits such a homeomorphism. However, most of the known examples of such continua have a complicated topological structure. For example, they are indecomposable (such as the pseudo-arc or the Knaster bucket-handle continuum), or they are not indecomposable but have some other complicated topological structure, such as a dense set of ramification points (such as the Sierpiński carpet) or a dense set of end-points (such as the Lelek fan). In this paper, we continue our mission of finding continua with simpler topological structures that admit a transitive homeomorphism. We construct a transitive homeomorphism on the Cantor fan. In our approach, we use two different techniques, each of them giving two constructions of a transitive homeomorphism on the Cantor fan: one technique using quotient spaces of products of compact metric spaces and Cantor sets, and one using Mahavier products of closed relations on compact metric spaces. We also demonstrate how our technique using Mahavier products of closed relations may be used to construct a transitive function f on a Cantor fan X such that $\varprojlim (X,f)$ is a Lelek fan.

We consider the system $\dot {x}=(A+ P(t,\epsilon )) x, x\in \mathbb {R}^{3}, $ where $A, P$ are both three-dimensional skew symmetric matrices, A is a constant matrix with eigenvalues $\pm i\unicode{x3bb} $ and 0, $P(t,\epsilon )$ is $C^{m}(m=0,1)$-smooth in $\epsilon $ and analytic quasi-periodic with respect to t with basic frequencies $\omega =(1,\alpha )$, with $\alpha $ being irrational, and $\epsilon $ is a small parameter. Under some non-resonant conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy condition, it is proved that for many sufficiently small parameters, this system can be reduced to a rotation system. Furthermore, if the basic frequencies satisfy that $ 0\leq \beta (\alpha ) < r,$ where $\beta (\alpha )=\limsup \nolimits _{n\rightarrow \infty } {\ln q_{n+1}}/{q_{n}},$ $q_{n}$ is the sequence of denominations of the best rational approximations for $\alpha \in \mathbb {R} \setminus \mathbb {Q},$ r is the initial radius of analytic domain, it is proved that for many sufficiently small parameters, this system can be reduced to a constant system.

In this article, we extend, with a great deal of generality, many results regarding the Hausdorff dimension of certain dynamical Diophantine coverings and shrinking target sets associated with a conformal iterated function system (IFS) previously established under the so-called open set condition. The novelty of the result we present is that it holds regardless of any separation assumption on the underlying IFS and thus extends to a large class of IFSs the previous results obtained by Beresnevitch and Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992] and by Barral and Seuret [The multifractal nature of heterogeneous sums of Dirac masses. Math. Proc. Cambridge Philos. Soc. 144(3) (2008), 707–727]. Moreover, it will be established that if S is conformal and satisfies mild separation assumptions (which are, for instance, satisfied for any self-similar IFS on $\mathbb {R}$ with algebraic parameters, no exact overlaps and similarity dimension smaller than $1$), then the classical result of Hill–Velani regarding the shrinking target problem associated with a conformal IFS satisfying the open set condition (and for which the Hausdorff measure was later computed by Allen and Barany [On the Hausdorff measure of shrinking target sets on self-conformal sets. Mathematika 67 (2021), 807–839]) can be extended.

We prove structural results for measure-preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions, these systems always have rational discrete spectrum and, as a consequence, zero entropy. We obtain several other refined structural and spectral results, one consequence of which is that the Archimedean characters are the only pretentious multiplicative functions that have Furstenberg systems with trivial rational spectrum, another is that a pretentious multiplicative function has ergodic Furstenberg systems if and only if it pretends to be a Dirichlet character, and a last one is that for any fixed pretentious multiplicative function, all its Furstenberg systems are isomorphic. We also study structural properties of Furstenberg systems of a class of multiplicative functions, introduced by Matomäki, Radziwiłł, and Tao, which lie in the intermediate zone between pretentiousness and strong aperiodicity. In a work of the last two authors and Gomilko, several examples of this class with exotic ergodic behavior were identified, and here we complement this study and discover some new unexpected phenomena. Lastly, we prove that Furstenberg systems of general bounded multiplicative functions have divisible spectrum. When these systems are obtained using logarithmic averages, we show that a trivial rational spectrum implies a strong dilation invariance property, called strong stationarity, but, quite surprisingly, this property fails when the systems are obtained using Cesàro averages.

We investigate and compare applications of the Zilber–Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way, we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu’s immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion.

This paper analyzes the initial value problem for the Toda lattice with almost periodic initial data: let $J(t; J_{0})$ denote the family of Jacobi matrices which are solutions of the Toda flow equation with initial condition $J(0; J_{0})=J_{0},$ then, the given almost periodic datum $J_{0}$ is a discrete linear Schrödinger operator with almost periodic potential, which plays a fundamental role in our considerations. We show that, under some given hypotheses, the spectrum of the Schrödinger operator is pure absolute continuous and homogeneous (measure-theoretically) by establishing exponential asymptotics on the size of spectral gaps. These two conclusions enable us to show the boundedness and almost periodicity in the time of solutions for Toda lattice equation with almost periodic initial data. As a consequence, our result presents a positive answer to the discrete Deift’s conjecture [Some open problems in random matrix theory and the theory of integrable systems. Integrable Systems and Random Matrices (Contemporary Mathematics, 458). American Mathematical Society, Providence, RI, 2008, pp. 419–430; Some open problems in random matrix theory and the theory of integrable systems. II. SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper no. 016].

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).

The rank of a tiling’s return module depends on the geometry of its tiles and is not a topological invariant. However, the rank of the first Čech cohomology $\check H^1(\Omega )$ gives upper and lower bounds for the rank of the return module. For all sufficiently large patches, the rank of the return module is at most the rank of $\check H^1(\Omega )$. For a generic choice of tile shapes and an arbitrary reference patch, the rank of the return module is at least the rank of $\check H^1(\Omega )$. Therefore, for generic tile shapes and all sufficiently large patches, the rank of the return module is equal to the rank of $\check H^1(\Omega )$.

Given a number field $\mathbb {K} \subset \mathbb {C}$ that is not contained in $\mathbb {R}$, we prove the existence of a dense set (with respect to the topology of local uniform convergence) of entire maps $f \colon \mathbb {C} \rightarrow \mathbb {C}$ whose preperiodic points and multipliers all lie in $\mathbb {K}$. This contrasts with the case of rational maps. In addition, we show that there exists an escaping quadratic-like map that is not conjugate to an affine escaping quadratic-like map and whose multipliers all lie in $\mathbb {Q}$.

Krieger’s embedding theorem provides necessary and sufficient conditions for an arbitrary subshift to embed in a given topologically mixing $\mathbb {Z}$-subshift of finite type. For certain families of $\mathbb {Z}^d$-subshifts of finite type, Lightwood characterized the aperiodic subsystems. In the current paper, we prove a new embedding theorem for a class of subshifts of finite type over any countable abelian group. Our theorem provides necessary and sufficient conditions for an arbitrary subshift X to embed inside a given subshift of finite type Y that satisfies a certain natural condition. For the particular case of $\mathbb {Z}$-subshifts, our new theorem coincides with Krieger’s theorem. Our result gives the first complete characterization of the subsystems of the multidimensional full shift $Y= \{0,1\}^{\mathbb {Z}^d}$. The natural condition on the target subshift Y, introduced explicitly for the first time in the current paper, is called the map extension property. It was introduced implicitly by Mike Boyle in the early 1980s for $\mathbb {Z}$-subshifts and is closely related to the notion of an absolute retract, introduced by Borsuk in the 1930s. A $\mathbb {Z}$-subshift has the map extension property if and only if it is a topologically mixing subshift of finite type. We show that various natural examples of $\mathbb {Z}^d$ subshifts of finite type satisfy the map extension property, and hence our embedding theorem applies for them. These include any subshift of finite type with a safe symbol and the k-colorings of $\mathbb {Z}^d$ with $k \ge 2d+1$. We also establish a new theorem regarding lower entropy factors of multidimensional subshifts that extends Boyle’s lower entropy factor theorem from the one-dimensional case.

We prove a result on equilibrium measures for potentials with summable variation on arbitrary subshifts over a countable amenable group. For finite configurations v and w, if v is always replaceable by w, we obtain a bound on the measure of v depending on the measure of w and a cocycle induced by the potential. We then use this result to show that under this replaceability condition, we can obtain bounds on the Lebesgue–Radon–Nikodym derivative $d (\mu _\phi \circ \xi ) / d\mu _\phi $ for certain holonomies $\xi $ that generate the homoclinic (Gibbs) relation. As corollaries, we obtain extensions of results by Meyerovitch [Gibbs and equilibrium measures for some families of subshifts. Ergod. Th. & Dynam. Sys. 33(3) (2013), 934–953], and García-Ramos and Pavlov [Extender sets and measures of maximal entropy for subshifts. J. Lond. Math. Soc. (2) 100(3) (2019), 1013–1033] to the countable amenable group subshift setting. Our methods rely on the exact tiling result for countable amenable groups by Downarowicz, Huczek, and Zhang [Tilings of amenable groups. J. Reine Angew. Math. 2019(747) (2019), 277–298] and an adapted proof technique from García-Ramos and Pavlov.

We combine methods from microlocal analysis and dimension theory to study resonances with largest real part for an Anosov flow with smooth real valued potential. We show that the resonant states are closely related to special systems of measures supported on the stable manifolds introduced by Climenhaga [SRB and equilibrium measures via dimension theory. A Vision for Dynamics in the 21st Century: The Legacy of Anatole Katok. Cambridge University Press, Cambridge, 2024, pp. 94–138]. As a result, we relate the presence of the resonances on the critical axis to mixing properties of the flow with respect to certain equilibrium measures and show that these equilibrium measures can be reconstructed from the spectral theory of the Anosov flow.

As discovered by W. Thurston, the action of a complex one-variable polynomial on its Julia set can be modeled by a geodesic lamination in the disk, provided that the Julia set is connected. It also turned out that the parameter space of such dynamical laminations of degree two gives a model for the bifurcation locus in the space of quadratic polynomials. This model is itself a geodesic lamination, the so called quadratic minor lamination of Thurston. In the same spirit, we consider the space of all cubic symmetric polynomials $f_\unicode{x3bb} (z)=z^3+\unicode{x3bb} ^2 z$ in three articles. In the first one, we construct the cubic symmetric comajor lamination together with the corresponding quotient space of the unit circle. As is verified in the third paper, this yields a monotone model of the cubic symmetric connectedness locus, that is, the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for generating the cubic symmetric comajor lamination analogous to the Lavaurs algorithm for constructing the quadratic minor lamination.

We show the existence of transcendental entire functions $f: \mathbb {C} \rightarrow \mathbb {C}$ with Hausdorff-dimension $1$ Julia sets, such that every Fatou component of f has infinite inner connectivity. We also show that there exist singleton complementary components of any Fatou component of f, answering a question of Rippon and Stallard [Eremenko points and the structure of the escaping set. Trans. Amer. Math. Soc. 372(5) (2019), 3083–3111]. Our proof relies on a quasiconformal-surgery approach developed by Burkart and Lazebnik [Interpolation of power mappings. Rev. Mat. Iberoam. 39(3) (2023), 1181–1200].

A cylindrical cascade on $\mathbb {T}^d\times \mathbb {R}^r$ can be seen as a deterministic random walk on $\mathbb {R}^r$ driven by an observable over the irrational toral translation on the base torus. We prove that, when the observable is the indicator function of a generic (straight) rectangle in $\mathbb {T}^2$, the cascade on $\mathbb {T}^2\times \mathbb {R}$ is ergodic for a $G_{\delta }$-dense set of translation vectors. We also provide examples of ergodic cylindrical cascades in higher dimensions with more restrictive conditions on the side lengths of the rectangles.

We study the class $\operatorname {Erg}^\perp $ of automorphisms which are disjoint with all ergodic systems. We prove that the identities are the only multipliers of $\operatorname {Erg}^\perp ,$ that is, each automorphism whose every joining with an element of $\operatorname {Erg}^{\perp }$ yields a system which is again an element of $\operatorname {Erg}^{\perp }$, must be an identity. Despite this fact, we show that $\operatorname {Erg}^\perp $ is closed by taking Cartesian products. Finally, we prove that there are non-identity elements in $\operatorname {Erg}^\perp $ whose self-joinings always yield elements in $\operatorname {Erg}^\perp $. This shows that there are non-trivial characteristic classes included in $\operatorname {Erg}^\perp $.