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Jon Chaika

Publications and source records attributed to Jon Chaika.

At least 37 records · Page 2Linked to original sources
arXiv2019

Self joinings of rigid rank one transformations arise as strong operator topology limits of convex combinations of powers

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 operator that is a strong operator topology limit of convex combinations of powers of the unitary operator given by the transformation.

math.DS↗
arXiv2018

Stationary coalescing walks on the lattice

We consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice. We assume that once walks meet, they coalesce. In $2d$, we classify the collective behavior of these walks under mild assumptions: they either coalesce almost surely or form bi-infinite trajectories. Bi-infinite trajectories form measure-preserving dynamical systems, have a common asymptotic direction in $2d$, and possess other nice properties. We use our theory to classify the behavior of compatible families of semi-infinite geodesics in stationary first- and last-passage percolation. We also partially answer a question raised by C. Hoffman about the limiting empirical measure of weights seen by geodesics. We construct several examples: our main example is a standard first-passage percolation model where geodesics coalesce almost surely, but have no asymptotic direction or average weight.

math.PR↗
arXiv2018

Self-Joinings for 3-IETs

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 Paulsen simplex.

math.DS↗
arXiv2018

A quantitative shrinking target result on Sturmian sequences for rotations

Let $R_α$ be an irrational rotation of the circle, and code the orbit of any point $x$ by whether $R_α^i(x)$ belongs to $[0,α)$ or $[α,1)$ -- this produces a Sturmian sequence. A point is undetermined at step $j$ if its coding up to time $j$ does not determine its coding at time $j+1$. We prove a pair of results on the asymptotic frequency of a point being undetermined, for full measure sets of $α$ and $x$.

math.DS↗
arXiv2018

Ergodicity of skew products over linearly recurrent IETs

We prove that the skew product over a linearly recurrent interval exchange transformation defined by almost any real-valued, mean-zero linear combination of characteristic functions of intervals is ergodic with respect to Lebesgue measure.

math.DS↗
arXiv2017

Logarithmic laws and unique ergodicity

We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodic properties of translation flow than hyperbolic geometry.

math.DS↗
arXiv2017

Circle averages and disjointness in typical flat surfaces on every Teichmueller disc

We prove that on the typical translation surface the flow in almost every pair of directions are not isomorphic to each other and are in fact disjoint. It was not known if there were any translation surfaces other than torus covers with this property. We provide an application to the convergence of `circle averages' for the flow (away from a sequence of radii of density 0) for such surfaces. Even the density of a sequence of 'circles' was only known in a few special examples.

math.DS↗
arXiv2015

Horocycle flow orbits and lattice surface characterizations

The orbit closure of any translation surface under the horocycle flow in almost any direction equals its $SL_2(\mathbb{R})$ orbit closure. This result gives rise to new characterizations of lattice surfaces in terms of the hororcycle flow.

math.DS↗
arXiv2015

Every flat surface is Birkhoff and Oseledets generic in almost every direction

We prove that the Birkhoff pointwise ergodic theorem and the Oseledets multiplicative ergodic theorem hold for every flat surface in almost every direction. The proofs rely on the strong law of large numbers, and on recent rigidity results for the action of the upper triangular subgroup of SL(2,R) on the moduli space of flat surfaces. Most of the results also use a theorem about continuity of splittings of the Kontsevich-Zorich cocycle recently proved by S. Filip.

math.DS↗