Rescaled Expansive Measures

• Rescaled expansive measures are a refinement of classical expansiveness that scales orbit separation by local flow speed, accommodating singularities in dynamical systems.
• They extend traditional fixed-radius methods by adapting error bounds dynamically to the local speed of the flow, ensuring the rescaled dynamical ball has zero measure.
• The theory connects entropy, hyperbolicity, and geometric stability, providing a robust framework for analyzing singular and multisingular hyperbolic flows.

Searching arXiv for papers on rescaled expansive measures, rescaled expansiveness for flows, and related measure-theoretic expansivity. Rescaled expansive measures are a measure-theoretic refinement of rescaled expansiveness for flows, designed to work robustly even in the presence of singularities. For a $C^1$ vector field $X$ on a compact manifold $M$ with flow $\phi^t$, the basic object is the rescaled dynamical ball

$$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$

where $$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$. A Borel probability measure $\mu$ is rescaled expansive if there exists $\varepsilon>0$ such that $\mu(\Gamma_\varepsilon(x))=0$ for every $x\in M$ (Yang, 22 Aug 2025). This notion arose from the earlier geometric notion of rescaling expansive flows, where orbit separation is measured relative to the local speed $X$0 rather than by a fixed metric radius (Wen et al., 2017).

1. Foundational rescaled expansiveness for flows

The geometric precursor of rescaled expansive measures is Wen–Wen’s rescaling expansiveness for flows. Let $X$1 be a compact Riemannian manifold, $X$2, $X$3 the generated flow, and $X$4 a compact invariant set. The flow is rescaling expansive on $X$5 if for any $X$6 there exists $X$7 such that, for any $X$8 and any increasing continuous $X$9,

$M$0

implies

$M$1

The crucial feature is the factor $M$2: when the flow slows down, especially near singularities, the allowed spatial error shrinks proportionally (Wen et al., 2017).

This rescaling is equivalent to classical expansiveness up to a constant factor for non-singular flows, because $M$3 is then bounded above and below. By contrast, for flows with singularities or very non-uniform speed, the rescaling is essential. Wen–Wen prove that every multisingular hyperbolic set, and therefore every singular hyperbolic set, is rescaling expansive. They also prove a generic converse: on a residual subset of $M$4, for any isolated chain transitive set $M$5, rescaling expansiveness, the local star property, and multisingular hyperbolicity are equivalent (Wen et al., 2017).

2. Measure-theoretic definition and reparametrization classes

The measure-theoretic formulation replaces orbit uniqueness by a zero-measure condition on the corresponding rescaled dynamical balls. In the 2025 formalization, the relevant reparametrization classes are

$M$6

$M$7

and, for $M$8,

$M$9

Using these classes, one defines

$\phi^t$0

and

$\phi^t$1

A Borel probability measure $\phi^t$2 is called rescaled expansive if there exists $\phi^t$3 such that $\phi^t$4 for every $\phi^t$5 (Yang, 22 Aug 2025).

A central structural theorem establishes equivalence among these formulations. For a flow generated by a $\phi^t$6 vector field on a compact manifold, the following are equivalent: $\phi^t$7 is rescaled expansive; for every $\phi^t$8 there exists $\phi^t$9 such that $$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$0 for $$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$1-almost every $$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$2; and for every $$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$3 there exists $$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$4 such that $$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$5 for $$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$6-almost every $$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$7 (Yang, 22 Aug 2025). The proof boosts continuous reparametrizations to increasing, almost identity reparametrizations by means of flowbox estimates and control of time distortion. For sufficiently small $$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$8, the corresponding dynamical balls coincide on regular points.

The forward-time analogue is also used. The positive rescaled dynamical ball is

$$\Gamma_\varepsilon(x) = \Big\{y \in M : \exists h \in C^0_0 \ \text{s.t.}\ d(\phi_s(x), \phi_{h(s)}(y))\leq \varepsilon \|X(\phi_s(x))\|\ \forall s\in \mathbb{R}\Big\},$$9

and a measure is positively rescaled expansive if $$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$0 for all $$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$1 and some $$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$2 (Yang, 22 Aug 2025).

3. Relation to earlier expansive measure theories

Rescaled expansive measures extend earlier measure-theoretic notions of expansiveness by replacing a fixed geometric scale with the local speed scale $$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$3. For homeomorphisms $$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$4, Morales’ measure expansiveness uses the dynamical ball

$$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$5

and requires $$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$6 for all $$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$7. Pacifico–Vieitez show that, on a residual subset of $$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$8, every Borel probability measure is expansive in this sense, while surface diffeomorphisms with homoclinic tangencies can be $$C^0_0=\{h:\mathbb{R}\to\mathbb{R}\text{ continuous with }h(0)=0\}$$9-approximated by non-measure expansive diffeomorphisms (Pacifico et al., 2013).

For flows, Carrasco–Morales introduced expansive measures via generalized dynamical balls built from continuous time reparametrizations. That theory yields several foundational properties: the support of an expansive measure avoids singularities, periodic points have measure zero, every orbit has measure zero, expansivity passes to time-$\mu$0 maps, and the notion is invariant under flow equivalence; suspension flows provide the natural bridge to the discrete-time theory (Carrasco-Olivera et al., 2013). The regular-flow refinement in 2025 replaces the original generalized dynamical balls by Borel dynamical balls and proves that every ergodic invariant measure with positive entropy is positively expansive (Pedrosa et al., 26 Jun 2025).

Against this background, the distinctive feature of rescaled expansive measures is the inequality

$\mu$1

which shrinks the allowable error near singularities and in regions of slow dynamics. This replaces absolute metric closeness by a local dynamical scale and thereby addresses a regime in which fixed-radius formulations are not dynamically natural.

4. Entropy, local entropy formula, and existence of invariant measures

The 2025 theory connects rescaled expansive measures directly to entropy. For regular $\mu$2, $\mu$3, and $\mu$4, three rescaled Bowen balls are introduced: $\mu$5

$\mu$6

$\mu$7

If $\mu$8 is an ergodic $\mu$9-invariant measure with $\varepsilon>0$0 and $\varepsilon>0$1, then $\varepsilon>0$2 is positively rescaled expansive; in particular, if $\varepsilon>0$3 has positive topological entropy, then it admits invariant rescaled expansive measures (Yang, 22 Aug 2025).

The same work proves a rescaled Brin–Katok local entropy formula. If $\varepsilon>0$4 is $\varepsilon>0$5-integrable and $\varepsilon>0$6 is ergodic and $\varepsilon>0$7-invariant, then for $\varepsilon>0$8-almost every $\varepsilon>0$9,

$\mu(\Gamma_\varepsilon(x))=0$0

for $\mu(\Gamma_\varepsilon(x))=0$1. This extends the fixed-point free flow result to general flows that may include singularities (Yang, 22 Aug 2025).

A complementary topological development assigns to every smooth vector field a rescaled topological entropy $\mu(\Gamma_\varepsilon(x))=0$2, defined through rescaled $\mu(\Gamma_\varepsilon(x))=0$3-spanning sets. This quantity is an upper bound for both the topological entropy and the rescaled metric entropy, coincides with the topological entropy for nonsingular vector fields, is positive for certain surface vector fields, is invariant under rescaled topological conjugacy, and bounds the growth rate of periodic orbits for rescaling expansive flows with dynamically isolated singular set (Rego et al., 3 Jun 2025). This situates rescaled expansive measures within a broader entropy theory adapted to singular flows.

5. Geometric mechanisms and hyperbolic settings

The proof of rescaling expansiveness for multisingular hyperbolic sets in Wen–Wen’s work is based on flowboxes with relative size, sectional Poincaré maps, and a discrete hyperbolic model. Around a regular point $\mu(\Gamma_\varepsilon(x))=0$4, the tangent box

$\mu(\Gamma_\varepsilon(x))=0$5

and the flowbox map

$\mu(\Gamma_\varepsilon(x))=0$6

admit uniform derivative bounds

$\mu(\Gamma_\varepsilon(x))=0$7

These relative estimates control the relation between spatial displacements and time differences at the local scale $\mu(\Gamma_\varepsilon(x))=0$8. In particular, if $\mu(\Gamma_\varepsilon(x))=0$9 and $x\in M$0 is small, then $x\in M$1. More generally, rescaled shadowing constrains reparametrizations to be almost translations (Wen et al., 2017).

Sampling at times $x\in M$2, one obtains sectional Poincaré maps $x\in M$3 on normal bundles. The multisingular hyperbolic structure provides domination and contraction/expansion after suitable cocycle reparametrization. The resulting discrete system has a unique bounded fixed point, forcing the normal displacement sequence to vanish. This yields the orbit-segment conclusion in the definition of rescaling expansiveness, and therefore proves that every multisingular hyperbolic set is rescaling expansive; every singular hyperbolic set is then rescaling expansive as a corollary (Wen et al., 2017).

A related development studies local rescaled stable and unstable sets for rescaled-expansive flows. For a non-singular compact invariant set $x\in M$4, rescaled expansiveness is characterized by

$x\in M$5

and non-trivial connected pieces of local rescaled stable and unstable sets appear at points that are neither $x\in M$6-stable nor $x\in M$7-unstable. If a rescaled-expansive flow admits a non-singular Lyapunov stable set that is not a finite union of compact orbits, then the flow has positive topological entropy (Arbieto et al., 2021). This suggests that the local geometry underlying rescaled expansive measures is closely tied to unstable continua and entropy production.

6. Scope, related developments, and open directions

Rescaled expansive measures belong to a broader family of measure-theoretic expansion notions, but they are not interchangeable with them. For measurable maps, the measure-theoretic expansion exponent $x\in M$8 is defined as the largest exponential rate such that almost every nearby point is expanded at least at rate $x\in M$9. Morales proves that $X$00, that a map expands small distances if and only if every Borel probability measure has positive expansion exponent, and that any nonatomic invariant measure with positive expansion exponent is positively expansive. For ergodic invariant measures, the Kolmogorov–Sinai entropy is bounded below by the product of the expansion exponent and the measure upper capacity (Morales, 23 Apr 2025). This is a quantitative theory for maps rather than flows, but it provides a natural comparison point.

For regular flows without fixed points, expansive measures admit an invariant-theoretic and descriptive-set-theoretic refinement. The Borel dynamical balls $X$01 yield a Brin–Katok local entropy formula for flows; every ergodic invariant measure with positive entropy is positively expansive; stable classes of such measures have zero measure; the set of expansive measures is a $X$02 subset in the weak* topology; and every expansive measure can be approximated by expansive measures supported on invariant sets (Pedrosa et al., 26 Jun 2025). This complements the rescaled theory by clarifying the nonsingular case.

Several open directions remain explicit. One question is whether there is a direct link between rescaling expansiveness and Bowen–Walters expansiveness for flows with singularities, possibly after passing to the natural extension $X$03 (Wen et al., 2017). Another group of questions concerns the measure-theoretic side: characterization of flows with positive entropy that admit rescaled expansive measures, relations with nonuniform hyperbolicity or SRB measures in singular flows, and extension of the rescaled Brin–Katok formula to non-ergodic measures (Yang, 22 Aug 2025). A plausible implication is that rescaled expansive measures will continue to serve as the natural measure-theoretic language whenever orbit separation must be evaluated relative to the local speed of the flow rather than at a fixed geometric scale.