There exists an interval exchange with a non-ergodic generic measure
Jon Chaika Howard Masur
Journal of Modern Dynamics 2015, 9(01): 289-304 doi: 10.3934/jmd.2015.9.289
We prove that there exists an interval exchange transformation and a point so that the orbit of the point equidistributes according to a non-ergodic measure. That is, it is possible for a non-ergodic measure to arise from the Krylov-Bogolyubov construction of invariant measures for an interval exchange transformation.
keywords: generic points Interval exchanges Rauzy induction.
Logarithmic laws and unique ergodicity
Jon Chaika Rodrigo Treviño
Journal of Modern Dynamics 2017, 11(1): 563-588 doi: 10.3934/jmd.2017022

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.

keywords: Unique ergodicity translation surfaces logarithm laws
A quantitative shrinking target result on Sturmian sequences for rotations
Jon Chaika David Constantine
Discrete & Continuous Dynamical Systems - A 2018, 38(10): 5189-5204 doi: 10.3934/dcds.2018229

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$.

keywords: Shrinking target Sturmian sequence circle rotation symbolic dynamics continued fractions
Hausdorff dimension for ergodic measures of interval exchange transformations
Jon Chaika
Journal of Modern Dynamics 2008, 2(3): 457-464 doi: 10.3934/jmd.2008.2.457
We show that there exist minimal interval-exchange transformations with an ergodic measure whose Hausdorff dimension is arbitrarily small, even 0. We will also show that in particular cases one can bound the Hausdorff dimension between $\frac{1}{2r+4}$ and $\frac{1}{r}$ for any r greater than 1.
keywords: non-unique ergodicity. Hausdorff dimension interval exchange transformation
Every flat surface is Birkhoff and Oseledets generic in almost every direction
Jon Chaika Alex Eskin
Journal of Modern Dynamics 2015, 9(01): 1-23 doi: 10.3934/jmd.2015.9.1
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 $SL(2,\mathbb R)$ 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.
keywords: flat surfaces. Ergodic theorems
Schrödinger operators defined by interval-exchange transformations
Jon Chaika David Damanik Helge Krüger
Journal of Modern Dynamics 2009, 3(2): 253-270 doi: 10.3934/jmd.2009.3.253
We discuss discrete one-dimensional Schrödinger operators whose potentials are generated by an invertible ergodic transformation of a compact metric space and a continuous real-valued sampling function. We pay particular attention to the case where the transformation is a minimal interval-exchange transformation. Results about the spectral type of these operators are established. In particular, we provide the first examples of transformations for which the associated Schrödinger operators have purely singular spectrum for every nonconstant continuous sampling function.
keywords: continuous spectrum singular spectrum interval-exchange transformations. Ergodic Schrödinger operators

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