Jon Chaika
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View article: Horocycle dynamics in rank one invariant subvarieties. I: Weak measure classification and equidistribution
Horocycle dynamics in rank one invariant subvarieties. I: Weak measure classification and equidistribution Open
Let ℳ be an invariant subvariety in the moduli space of translation surfaces. We contribute to the study of the dynamical properties of the horocycle flow on ℳ . In the context of dynamics on the moduli space of translation surfaces, we in…
View article: Veech Surfaces and Expanding Twist Tori on Moduli Spaces of Abelian Differentials
Veech Surfaces and Expanding Twist Tori on Moduli Spaces of Abelian Differentials Open
Let $(M,ω)$ be a translation surface such that every leaf of its horizontal foliation is either closed, or joins two zeros of $ω$. Then, $M$ decomposes as a union of horizontal Euclidean cylinders. The $\textit{twist torus}$ of $(M,ω)$, de…
View article: A rank one mild mixing system without minimal self joinings
A rank one mild mixing system without minimal self joinings Open
We show that there is a rank 1 transformation that is mildly mixing but does not have minimal self-joinings, answering a question of Thouvenot.
View article: Path-connectivity of Thick Laminations, and Markov Processes with Thick Limit Sets
Path-connectivity of Thick Laminations, and Markov Processes with Thick Limit Sets Open
A lamination $λ$ is $ε$-thick (with respect to a basepoint $X$), if the Teichmüller ray from $X$ in the direction of $λ$ stays in the $ε$-thick part. We show that, for surfaces of high enough genus, any two $ε$-thick laminations can be joi…
View article: Weak mixing in rational billiards
Weak mixing in rational billiards Open
We completely characterize rational polygons whose billiard flow is weakly mixing in almost every direction as those which are not almost integrable, in the terminology of Gutkin, modulo some low complexity exceptions. This proves a longst…
View article: Shrinking rates of horizontal gaps for generic translation surfaces
Shrinking rates of horizontal gaps for generic translation surfaces Open
A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface invol…
View article: On the ergodic theory of the real Rel foliation
On the ergodic theory of the real Rel foliation Open
Let ${{\mathcal {H}}}$ be a stratum of translation surfaces with at least two singularities, let $m_{{{\mathcal {H}}}}$ denote the Masur-Veech measure on ${{\mathcal {H}}}$ , and let $Z_0$ be a flow on $({{\mathcal {H}}}, m_{{{\mathcal {H}…
View article: The horocycle flow on the moduli space of translation surfaces
The horocycle flow on the moduli space of translation surfaces Open
We survey some results on the dynamics of the horocycle flow on the moduli space of translation surfaces. We outline proofs of some recent results, obtained by the authors in collaboration with John Smillie, and pose some open questions.
View article: Connectivity of the Gromov Boundary of the Free Factor Complex
Connectivity of the Gromov Boundary of the Free Factor Complex Open
We show that in large enough rank, the Gromov boundary of the free factor complex is path connected and locally path connected.
View article: Horocycle dynamics in rank one invariant subvarieties I: weak measure classification and equidistribution
Horocycle dynamics in rank one invariant subvarieties I: weak measure classification and equidistribution Open
Let M be an invariant subvariety in the moduli space of translation surfaces. We contribute to the study of the dynamical properties of the horocycle flow on M. In the context of dynamics on the moduli space of translation surfaces, we int…
View article: On the ergodic theory of the real Rel foliation
On the ergodic theory of the real Rel foliation Open
Let $\mathcal{H}$ be a stratum of translation surfaces with at least two singularities, let $m_{\mathcal{H}}$ denote the Masur-Veech measure on $\mathcal{H}$, and let $Z_0$ be a flow on $(\mathcal{H}, m_{\mathcal{H}})$ obtained by integrat…
View article: Pairs in discrete lattice orbits with applications to Veech surfaces
Pairs in discrete lattice orbits with applications to Veech surfaces Open
Let $Λ_1$, $Λ_2$ be two discrete orbits under the linear action of a lattice $Γ<\mathrm{SL}_2(\mathbb{R})$ on the Euclidean plane. We prove a Siegel$-$Veech-type integral formula for the averages $$ \sum_{\mathbf{x}\inΛ_1} \sum_{\mathbf{y}…
View article: Zero measure spectrum for multi-frequency Schrödinger operators
Zero measure spectrum for multi-frequency Schrödinger operators Open
Building on works of Berthé–Steiner–Thuswaldner and Fogg–Nous, we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As …
View article: Shrinking rates of horizontal gaps for generic translation surfaces
Shrinking rates of horizontal gaps for generic translation surfaces Open
A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface invol…
View article: Stationary coalescing walks on the lattice II: entropy
Stationary coalescing walks on the lattice II: entropy Open
This paper is a sequel to Chaika and Krishnan, 2016. We again consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice . We assume that once walks meet, they coalesce. We cons…
View article: Connectivity of the Gromov Boundary of the Free Factor Complex
Connectivity of the Gromov Boundary of the Free Factor Complex Open
We show that in large enough rank, the Gromov boundary of the free factor complex is path connected and locally path connected.
View article: Self-joinings for 3-IETs
Self-joinings for 3-IETs Open
We show that typical interval exchange transformations on three intervals are not 2-simple answering a question of Veech. Moreover, the set of self-joinings of almost every 3-IET is a Poulsen simplex.
View article: On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials
On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials Open
We study the horocycle flow on the stratum of translation surfaces $\mathcal{H}(2)$. We show that there is a sequence of horocycle ergodic measures, each supported on a periodic horocycle orbit, which weakly converges to an invariant, but …
View article: A prime system with many self-joinings
A prime system with many self-joinings Open
We construct a rigid, rank 1, prime transformation that is not quasi-simple and whose self-joinings form a Paulsen simplex. This seems to be the first example of a prime system whose self-joinings form a Paulsen simplex.
View article: Zero Measure Spectrum for Multi-Frequency Schrödinger Operators
Zero Measure Spectrum for Multi-Frequency Schrödinger Operators Open
Building on works of Berthé--Steiner--Thuswaldner and Fogg--Nous we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. A…
View article: Tremors and horocycle dynamics on the moduli space of translation surfaces
Tremors and horocycle dynamics on the moduli space of translation surfaces Open
We introduce a "tremor" deformation on strata of translation surfaces. Using it, we give new examples of behaviors of horocycle flow orbits in strata of translation surfaces. In the genus two stratum with two singular points, we find orbit…
View article: Weakly Mixing Polygonal Billiards
Weakly Mixing Polygonal Billiards Open
We prove that there exists a residual set of (non-rational) polygons such the billiard flow is weakly mixing with respect to the Liouville measure (on the unit tangent bundle to the billiard). This follows, via a Baire category argument, f…
View article: Path-connectivity of the set of uniquely ergodic and cobounded foliations
Path-connectivity of the set of uniquely ergodic and cobounded foliations Open
We show that for a closed surface of genus at least 5, or a surface of genus at least 2 with at least one marked point, the set of uniquely ergodic foliations and the set of cobounded foliations is path-connected and locally path-connected.
View article: Self joinings of rigid rank one transformations arise as strong operator topology limits of convex combinations of powers
Self joinings of rigid rank one transformations arise as strong operator topology limits of convex combinations of powers Open
This is a straightforward generalization Section 2 of arXiv:1805.11167. It shows that for a residual set of transformations in the space of measure preserving transformations, with the weak topology, any self-joining defines a Markov opera…
View article: Möbius disjointness for interval exchange transformations on three intervals
Möbius disjointness for interval exchange transformations on three intervals Open
We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.
View article: Uniform distribution of saddle connection lengths (with an appendix by Daniel El-Baz and Bingrong Huang)
Uniform distribution of saddle connection lengths (with an appendix by Daniel El-Baz and Bingrong Huang) Open
For any SL(2,R) invariant and ergodic probability measure on any stratum of flat surfaces, almost every flat surface has the property that its non-decreasing sequence of saddle connection lengths is uniformly distributed mod one.
View article: The typical measure preserving transformation is not an interval\n exchange transformation
The typical measure preserving transformation is not an interval\n exchange transformation Open
We show that the typical measure preserving transformation is not isomorphic\nto any interval exchange transformation.\n
View article: The typical measure preserving transformation is not an interval exchange transformation
The typical measure preserving transformation is not an interval exchange transformation Open
We show that the typical measure preserving transformation is not isomorphic to any interval exchange transformation.