arXiv subjects

Omri Nisan Solan

Publications and source records attributed to Omri Nisan Solan.

11 recordsLinked to original sources

arXiv2025

Tori Approximation of Families of Diagonally Invariant Measures

We approximate any portion of any orbit of the full diagonal group $A$ in the space of unimodular lattices in $\RR^n$ using a fixed proportion of a compact $A$-orbit. Using those approximations for the appropriate sequence of orbits, we prove the existence of non-ergodic measures which are also weak limits of compactly supported $A$-invariant measures. In fact, given any countably many $A$-invariant ergodic measures, our methods show that there exists a sequence of compactly supported periodic $A$-invariant measures such that the ergodic decomposition of its weak limit has these measures as factors with positive weight. Using the same methods, we prove that any compactly supported $A$-invariant and ergodic measure is the weak limit of the restriction of different compactly supported periodic measures to a fixed proportion of the time. In addition, for any $c\in (0,1]$ we find a sequence of compactly supported periodic $A$-invariant measures that converge weakly to $cm_{X_n}$ where $m_{X_n}$ denotes the Haar measure on $X_n$. In particular, we prove the existence of partial escape of mass for compact $A$-orbits. These results give affirmative answers to questions posed by Shapira in ~\cite{ShapiraEscape}. Our proofs are based on a modification of Shapira's proof in ~\cite{ShapiraEscape} and on a generalization of a construction of Cassels, as well as on effective equidistribution estimates of Hecke neighbors by Clozel, Oh and Ullmo, and a number theoretic construction of a special number field.

math.DS↗

arXiv2024

Birkhoff generic points on curves in horospheres

Let $\{a_t: t \in \mathbb{R}\}< SL_{d}(\mathbb{R})$ be a diagonalizable subgroup whose expanding horospherical subgroup $U < SL_{d}(\mathbb{R})$ is abelian. By the Birkhoff ergodic theorem, for any $x \in SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$ and for almost every point $u \in U$ the point $ux$ is Birkhoff generic for $a_t$ when $t \to \infty$. We prove that the same is true when $U$ is replaced by any non-degenerate analytic curve in $U$. This Birkhoff genericity result has various applications in Diophantine approximation. For instance, we obtain density estimates for Dirichlet improvability along typical points on a curve in Euclidean space. Other applications address approximations by algebraic numbers and best approximations (in the sense of Lagarias).

math.DS↗

arXiv2024

Critical exponent gap and leafwise dimension

We show that for every nonarithmetic lattice $Γ<{\rm SL}_2(\mathbb{C})$ there is a gap $\varepsilon_Γ>0$ such that for every $g\in {\rm SL}_2(\mathbb{C})$ the intersection ${\rm SL}_2(\mathbb{R})\cap gΓg^{-1}$ is either a lattice in ${\rm SL}_2(\mathbb{R})$ or has critical exponent $δ({\rm SL}_2(\mathbb{R})\cap gΓg^{-1}) \leq 1 - \varepsilon_Γ$.

math.DS↗

arXiv2024

Orbits in Teichmüller dynamics admits a critical exponent gap

McMullen '03 constructs a collection of orbits $\mathrm{SL}_2(\mathbb{R}).x$ in $\mathcal{H}(1,1)$ with infinitely generated stabilizers $\mathrm{stab}_{\mathrm{SL}_2(\mathbb{R})}(x)$. We prove a gap in the set of critical exponents of stabilizers of $\mathrm{SL}_2(\mathbb{R})$-orbits in $\mathcal{H}_g$: for every $x\in \mathcal{H}_g$, either $\mathrm{stab}_{\mathrm{SL}_2(\mathbb{R})}(x)$ is a lattice, or we have a uniform bound on the critical exponent $δ(\mathrm{stab}_{\mathrm{SL}_2(\mathbb{R})}(x)) \le 1-\varepsilon_g$.

math.DS↗

arXiv2024

Automatic continuity of Polynomial maps and cocycles

Classical theorems from the early 20th century state that any Haar measurable homomorphism between locally compact groups is continuous. In particular, any Lebesgue-measurable homomorphism $ϕ:\mathbb{R} \to \mathbb{R}$ is of the form $ϕ(x)=ax$ for some $a \in \mathbb{R}$. In this short note, we prove that any Lebesgue measurable function $ϕ:\mathbb{R} \to \mathbb{R}$ that vanishes under any $d+1$ ``difference operators'' is a polynomial of degree at most $d$. More generally, we prove the continuity of any Haar measurable polynomial map between locally compact groups, in the sense of Leibman. We deduce the above result as a direct consequence of a theorem about the automatic continuity of cocycles.

math.GT↗

arXiv2022

Browder's Theorem: from One-Dimensional Parameter Space to General Parameter Space

A parametric version of Brouwer's Fixed Point Theorem, which is proven using the fixed-point index, states that for every continuous mapping $f : (X \times Y) \to Y$, where $X$ is nonempty, compact, and connected subset of a Hausdorff topological space and $Y$ is a nonempty, convex, and compact subset of a locally-convex topological vector space, the set of fixed points of $f$, defined by $C_f := \{ (x,y) \in X \times Y \colon f(x,y)=y\}$, has a connected component whose projection onto the first coordinate is $X$. In this note we provide an elementary proof for this result, using its reduction to the case $X = [0,1]$.

math.GN↗

arXiv2021

Parametric Geometry of Numbers with General Flow

We extend results of parametric geometry of numbers to a general diagonal flow on the space of lattices. Moreover, we compute the Hausdorff dimension of the set of trajectories with every given behavior, with respect to a nonstandard metric and thereby attain bounds on the standard ones.

math.DS↗

arXiv2021

Browder's Theorem with General Parameter Space

Browder (1960) proved that for every continuous function $F : X \times Y \to Y$, where $X$ is the unit interval and $Y$ is a nonempty, convex, and compact subset of $\dR^n$, the set of fixed points of $F$, defined by $C_F := \{ (x,y) \in X \times Y \colon F(x,y)=y\}$ has a connected component whose projection to the first coordinate is $X$. We extend this result to the case where $X$ is a connected and compact Hausdorff space.

math.GN↗

arXiv2019

Sunspot Equilibrium in General Quitting Games

We prove that positive recursive general quitting games, which are quitting games in which each player may have more than one continue action, admit a sunspot $\ep$-equilibrium, for every $\ep > 0$. To this end we show that the equilibrium set of strategic-form games can be uniformly approximated by a smooth manifold, and develop a new fixed-point theorem for smooth manifolds.

math.PR↗

arXiv2018

Jointly Controlled Lotteries with Biased Coins

We provide a mechanism that uses two biased coins and implements any distribution on a finite set of elements, in such a way that even if the outcomes of one of the coins is determined by an adversary, the final distribution remains unchanged. We apply this result to show that every quitting game in which at least two players have at least two continue actions has an undiscounted $\ep$-equilibrium, for every $\ep > 0$.

math.PR↗

arXiv2016

Monovex Sets

A set $A$ in a finite dimensional Euclidean space is \emph{monovex} if for every two points $x,y \in A$ there is a continuous path within the set that connects $x$ and $y$ and is monotone (nonincreasing or nondecreasing) in each coordinate. We prove that every open monovex set as well as every closed monovex set is contractible, and provide an example of a nonopen and nonclosed monovex set that is not contractible. Our proofs reveal additional properties of monovex sets.

math.GN↗