arXiv subjects

Marlies Gerber

Publications and source records attributed to Marlies Gerber.

8 recordsLinked to original sources

arXiv2025

Non-classifiability of mixing zero-entropy diffeomorphisms up to isomorphism

We show that the problem of classifying, up to isomorphism, the collection of zero-entropy mixing automorphisms of a standard non-atomic probability space, is intractible. More precisely, the collection of isomorphic pairs of automorphisms in this class is not Borel, when considered as a subset of the Cartesian product of the collection of measure-preserving automorphisms with itself. This remains true if we restrict to zero-entropy mixing automorphisms that are also $C^{\infty}$ diffeomorphisms of the five-dimensional torus. In addition, both of these results still hold if ``isomorphism'' is replaced by ``Kakutani equivalence.'' In our argument we show that for a uniquely and totally ergodic automorphism $U$ and a particular family of automorphisms $\mathcal{S}$, if $T\times U$ is isomorphic to $T^{-1}\times U$ with $T\in\mathcal{S}$ then $T$ is isomorphic to ${T^{-1}}$. However, this type of ``cancellation'' of factors from isomorphic Cartesian products is not true in general. We present an example due to M. Lemańczyk of two weakly mixing automorphisms $T$ and $S$ and an irrational rotation $R$ such that $T\times R$ is isomorphic to $S\times R$, but $T$ and $S$ are not isomorphic.

math.DS↗

arXiv2024

Non-classifiability of Ergodic Flows up to Time Change

A time change of a flow $\{T_t\}$, ${t\in\mathbb{R}}$, is a reparametrization of the orbits of the flow such that each orbit is mapped to itself by an orientation-preserving homeomorphism of the parameter space. If a flow $\{S_t\}$ is isomorphic to a flow obtained by a reparametrization of a flow $\{T_t\}$, then we say that $\{S_t\}$ and $\{T_t\}$ are isomorphic up to a time change. For ergodic flows $\{S_t\}$ and $\{T_t\}$, Kakutani showed that this happens if and only if the two flows have Kakutani equivalent transformations as cross-sections. We prove that the Kakutani equivalence relation on ergodic invertible measure-preserving transformations of a standard non-atomic probability space is not a Borel set. This shows in a precise way that classification of ergodic transformations up to Kakutani equivalence is impossible. In particular, our results imply the non-classifiability of ergodic flows up to isomorphism after a time change. Moreover, we obtain anti-classification results under isomorphism for ergodic invertible transformations of a sigma-finite measure space. We also obtain anti-classification results under Kakutani equivalence for ergodic area-preserving smooth diffeomorphisms of the disk, annulus, and 2-torus, as well as real-analytic diffeomorphisms of the $2$-torus. Our work generalizes the anti-classification results under isomorphism for ergodic transformations obtained by Foreman, Rudolph, and Weiss.

math.DS↗

arXiv2023

Non-Classifiability of Kolmogorov Diffeomorphisms up to Isomorphism

We consider the problem of classifying Kolmogorov automorphisms (or $K$-automorphisms for brevity) up to isomorphism. Within the collection of measure-preserving transformations, Bernoulli shifts have the ultimate mixing property, and $K$-automorphisms have the next-strongest mixing properties of any widely considered family of transformations. J. Feldman observed that unlike Bernoulli shifts, the family of $K$-automorphisms cannot be classified up to isomorphism by a complete numerical Borel invariant. This left open the possibility of classifying $K$-automorphisms with a more complex type of Borel invariant. We show that this is impossible, by proving that the isomorphism equivalence relation restricted to $K$-automorphisms, considered as a subset of the Cartesian product of the set of $K$-automorphisms with itself, is a complete analytic set, and hence not Borel. Moreover, we prove this remains true if we restrict consideration to $K$-automorphisms that are also $C^{\infty}$ diffeomorphisms. This shows in a concrete way that the problem of classifying $K$-automorphisms up to isomorphism is intractible.

math.DS↗

arXiv2019

Loosely Bernoulli Odometer-Based Systems Whose Corresponding Circular Systems Are Not Loosely Bernoulli

M. Foreman and B. Weiss obtained an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphism, by using a functor $\mathcal{F}$ mapping odometer-based systems, $\mathcal{OB}$, to circular systems, $\mathcal{CB}$. This functor transfers the classification problem from $\mathcal{OB}$ to $\mathcal{CB}$, and it preserves weakly mixing extensions, compact extensions, factor maps, the rank-one property, and certain types of isomorphisms. Thus it is natural to ask whether $\mathcal{F}$ preserves other dynamical properties. We show that $\mathcal{F}$ does not preserve the loosely Bernoulli property by providing positive and zero entropy examples of loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli. We also construct a loosely Bernoulli circular system whose corresponding odometer-based system has zero entropy and is not loosely Bernoulli.

math.DS↗

arXiv2018

A smooth zero-entropy diffeomorphism whose product with itself is loosely Bernoulli

Let $M$ be a smooth compact connected manifold of dimension $d\geq 2$, possibly with boundary, that admits a smooth effective $\mathbb{T}^2$-action $\mathcal{S}=\left\{S_{α,β}\right\}_{(α,β) \in \mathbb{T}^2}$ preserving a smooth volume $ν$, and let $\mathcal{B}$ be the $C^{\infty}$ closure of $\left\{h \circ S_{α,β} \circ h^{-1} \;:\;h \in \text{Diff}^{\infty}\left(M,ν\right), (α,β) \in \mathbb{T}^2\right\}$. We construct a $C^{\infty}$ diffeomorphism $T \in \mathcal{B}$ with topological entropy $0$ such that $T \times T$ is loosely Bernoulli. Moreover, we show that the set of such $T \in \mathcal{B}$ contains a dense $G_δ$ subset of $\mathcal{B}$. The proofs are based on a two-dimensional version of the approximation-by-conjugation method.

math.DS↗

arXiv2015

Generic Absence of Finite Blocking for Interior Points of Birkhoff Billiards

Let x and y be points in a billiard table M that is bounded by a curve sigma. We assume that sigma is a simple closed C^r curve with positive curvature, where r is at least 2. A subset B of M\{x,y} is called a blocking set for the pair (x,y) if every billiard path in M from x to y passes through a point in B. If a finite blocking set exists, the pair (x,y) is called secure in M; if not, it is called insecure. We show that for the generic (in the sense of Baire category) curve sigma, the generic pair of interior points is insecure.

math.DS↗

arXiv2011

Real Analytic Metrics on S^2 with Total Absence of Finite Blocking

If (M,g) is a Riemannian manifold and x,y are points in M, then a subset P of M\{x,y} is said to be a blocking set for (x,y) if every geodesic from x to y passes through a point of P. If no pair (x,y) in M X M has a finite blocking set, then (M,g) is said to be totally insecure. We prove that there exist real analytic metrics h on S^2 such that (S^2,h) is totally insecure.

math.DG↗

arXiv2010

A dense G-delta set of Riemannian metrics without the finite blocking property

A pair of points (x,y) in a Riemannian manifold (M,g) is said to have the finite blocking property if there is a finite set P contained in M\{x,y} such that every geodesic segment from x to y passes through a point of P. We show that for every closed C-infinity manifold M of dimension at least two and every pair (x,y) in M x M, there exists a dense G-delta set of C-infinity Riemannian metrics on M such that (x,y) fails to have the finite blocking property for every g in that set.

math.DG↗