[Paper Review] Lebesgue Orbit Equivalence of Multidimensional Borel Flows

Konstantin Slutsky|arXiv (Cornell University)|Apr 3, 2015

Mathematical Dynamics and Fractals6 references3 citations

TL;DR

This paper establishes that two free multidimensional Borel flows of $\mathbb{R}^d$ are Lebesgue Orbit Equivalent (LOE) if and only if they have the same number of invariant ergodic probability measures. The proof uses a back-and-forth construction with measure-preserving maps on Borel rectangles, leveraging cocompact cross sections and compressible equivalence relations to extend orbit equivalences while preserving Lebesgue measure on each orbit.

ABSTRACT

The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue Orbit Equivalence, by which we understand an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non smooth Euclidean flows are shown to be Lebesgue Orbit Equivalence if and only if they admit the same number of invariant ergodic probability measures.

Motivation & Objective

• To classify free multidimensional Borel flows of $\mathbb{R}^d$ up to Lebesgue Orbit Equivalence (LOE), a strengthening of orbit equivalence that preserves Lebesgue measure on each orbit.
• To determine whether the cardinality of the set of invariant ergodic probability measures (pie measures) serves as a complete invariant for LOE in the context of non-smooth free $\mathbb{R}^d$-flows.
• To extend orbit equivalences across orbits in a measure-preserving way using a Borel back-and-forth argument, even when the full phase space is not smooth.
• To overcome the obstruction that LOE does not preserve Haar measure in the same way as discrete group actions, by focusing on Lebesgue measure on orbits rather than global measures.

Proposed method

• Construct a Borel isomorphism $\phi: X \to Y$ between two free $\mathbb{R}^d$-flows using a back-and-forth method on Borel rectangles $B^k$ and cross sections $\mathcal{C}^k$.
• Use cocompact cross sections $\mathcal{C}_X \subseteq X$ and $\mathcal{C}_Y \subseteq Y$ to define a base structure for the orbit equivalence, ensuring uniform full measure in the limit.
• Define measure-preserving maps on $B^k$ blocks by transferring Lebesgue measure via the flow action, ensuring that $\phi$ preserves Lebesgue measure on each orbit.
• Apply Theorem 8.1 to extend the orbit equivalence across $B^k$ blocks in a way that maintains measure preservation and Borel measurability at each stage.
• Use compressible, non-smooth equivalence relations on cross sections to construct invariant subsets $X_0 \subseteq X$ and $Y_0 \subseteq Y$ with no pie measures, enabling the extension of the equivalence to the full space.
• Combine the results of Theorem 7.2 (existence of LOE on sets of uniformly full measure) with Theorem 8.1 (extension to full flows) to construct a global LOE between the original flows.

Experimental results

Research questions

• RQ1 Is the number of invariant ergodic probability measures a complete invariant for Lebesgue Orbit Equivalence of free multidimensional Borel flows?
• RQ2 Can a Borel orbit equivalence between free $\mathbb{R}^d$-flows be extended to a Lebesgue measure-preserving orbit equivalence on all orbits?
• RQ3 How can one construct a LOE when the phase space is not smooth and contains no invariant probability measures?
• RQ4 To what extent does the failure of orbit equivalence to preserve Haar measure in continuous group actions affect classification in the Borel setting?
• RQ5 Can the back-and-forth method be adapted to preserve Lebesgue measure on orbits while maintaining Borel measurability in the context of multidimensional flows?

Key findings

• Two free non-smooth $\mathbb{R}^d$-flows are Lebesgue Orbit Equivalent if and only if they have the same number of invariant ergodic probability measures.
• The existence of a LOE between two such flows is guaranteed whenever the cardinalities of their sets of pie measures coincide.
• The construction of the LOE proceeds via a Borel back-and-forth argument on Borel rectangles $B^k$, ensuring measure preservation on each orbit through careful transfer of Lebesgue measure.
• The method successfully extends a LOE defined on a full-measure subset to the entire space, even when the complement is non-smooth and has no invariant measures.
• The use of compressible, non-smooth equivalence relations on cocompact cross sections allows the construction of invariant subsets with no pie measures, enabling the extension of the equivalence.
• The result generalizes the classical Dougherty–Jackson–Kechris classification to the continuous, measure-preserving setting of multidimensional flows, showing that the number of pie measures is the sole invariant under LOE.