Conservative IID Random Dynamical Systems

• Conservative IID random dynamical systems are defined as independent, identically distributed volume-preserving maps on a finite-volume manifold, key to understanding non-uniform hyperbolicity.
• These systems employ shift-space and skew-product formulations to study the nonvanishing of extremal Lyapunov exponents and the rigidity in zero-exponent regimes.
• Coexpansion on average in this framework underpins spectral gaps in annealed transfer operators, leading to exponential mixing and central limit theorems.

Conservative IID random dynamical systems are random compositions of independent, identically distributed volume-preserving maps on a finite-volume or closed Riemannian manifold. In the formulations considered in recent work, the randomness is encoded by a shift space and a skew-product, while the conservative structure means that each sampled map preserves the reference volume measure, so the product or stationary measure is preserved by the random evolution. This framework supports several distinct but closely related lines of analysis: generic nonvanishing of extremal Lyapunov exponents, invariance-principle rigidity when exponents vanish, spectral-gap mechanisms for annealed transfer operators, and implications from annealed statistical laws of the two-point motion to quenched laws for the one-point motion (Barrientos et al., 2018, DeWitt et al., 10 Mar 2025, DeWitt et al., 5 Sep 2025, Magno et al., 13 Jan 2026).

1. Formal setup and basic notions

A standard finite-support model fixes a connected, finite-volume Riemannian manifold $M$ of dimension $d\ge 2$, endowed with its normalized Lebesgue measure $m$, an integer $k\ge 2$, and weights $p_1,\dots,p_k>0$ with $\sum_i p_i=1$. One then defines

$$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$

lets $\theta:\Omega\to\Omega$ be the left shift, associates to each symbol $i$ a $C^r$-diffeomorphism $d\ge 2$0 preserving $d\ge 2$1, and forms the skew-product

$d\ge 2$2

This random transformation is invertible, preserves $d\ge 2$3, and is often assumed ergodic. Equivalently, the group generated by $d\ge 2$4 acts ergodically on $d\ge 2$5, meaning that any measurable set invariant under all $d\ge 2$6 has either zero or full $d\ge 2$7-measure (Barrientos et al., 2018).

A measure-driven version replaces the finite alphabet by a probability measure $d\ge 2$8 or $d\ge 2$9 on a space of conservative maps. One formulation takes a closed smooth Riemannian manifold $m$0 with volume form $m$1, a probability measure $m$2 on $m$3 with compact support, and an IID sequence $m$4 chosen with law $m$5. Equivalently, one works on a product space $m$6 with shift $m$7 and skew-product

$m$8

Another formulation uses a standard Borel probability space $m$9, the one-sided shift $k\ge 2$0, and a measurable map $k\ge 2$1, $k\ge 2$2, such that for $k\ge 2$3-almost every $k\ge 2$4, the map $k\ge 2$5 is volume-preserving and piecewise $k\ge 2$6 on a full-measure domain. In all of these formulations, conservativity means that each sampled map preserves the reference volume, and the resulting volume is stationary under the random dynamics (DeWitt et al., 10 Mar 2025, DeWitt et al., 5 Sep 2025, Magno et al., 13 Jan 2026).

Two derived motions are especially important. The one-point motion is the random composition $k\ge 2$7 acting on $k\ge 2$8. The two-point motion is the induced map

$k\ge 2$9

on $p_1,\dots,p_k>0$0, which again preserves the product volume. This distinction becomes central in the comparison between annealed and quenched statistical behavior (DeWitt et al., 5 Sep 2025).

2. Lyapunov exponents and generic non-uniform hyperbolicity

For random products of conservative diffeomorphisms, the extremal Lyapunov exponents are defined through the random Oseledets theorem. For $p_1,\dots,p_k>0$1-almost every $p_1,\dots,p_k>0$2,

$p_1,\dots,p_k>0$3

and

$p_1,\dots,p_k>0$4

Ergodicity of $p_1,\dots,p_k>0$5 under $p_1,\dots,p_k>0$6 implies that these are constant $p_1,\dots,p_k>0$7-almost everywhere, written $p_1,\dots,p_k>0$8 and $p_1,\dots,p_k>0$9. Non-uniform hyperbolicity means

$\sum_i p_i=1$0

In the piecewise $\sum_i p_i=1$1 setting, one likewise obtains extremal exponents $\sum_i p_i=1$2 and $\sum_i p_i=1$3 for $\sum_i p_i=1$4-almost every $\sum_i p_i=1$5 (Barrientos et al., 2018, Magno et al., 13 Jan 2026).

A central genericity result states that, after fixing $\sum_i p_i=1$6, $\sum_i p_i=1$7, and conservative diffeomorphisms $\sum_i p_i=1$8 such that $\sum_i p_i=1$9 is ergodic under the group $$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$0, there exists a set

$$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$1

which is $$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$2-open and $$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$3-dense, such that for every $$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$4 the IFS generated by $$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$5 satisfies

$$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$6

Thus none of the extremal Lyapunov exponents vanishes (Barrientos et al., 2018).

In dimension $$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$7, this has a stronger formulation: the set of non-uniformly hyperbolic ergodic IFSs of independent volume-preserving $$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$8-diffeomorphisms contains a $$\Omega=\{1,\dots,k\}^{\mathbb Z}, \qquad \mathbb P=p^{\mathbb Z},$$9-open, $\theta:\Omega\to\Omega$0-dense subset. In higher dimensions, the result ensures nonzero extremal exponents, though intermediate exponents may vanish. The same source emphasizes that this contrasts sharply with the deterministic conservative case, where robust zero-exponent phenomena are known; here IID randomness restores a form of hyperbolicity generically (Barrientos et al., 2018).

3. Invariance principles and rigidity of the zero-exponent regime

A recurring structural theme is that vanishing or coincident extremal Lyapunov exponents force invariant measures on the projective bundle. In the setting of random products of conservative diffeomorphisms, if $\theta:\Omega\to\Omega$1, an invariance principle due to Ledrappier, Crauel, and Viana yields a $\theta:\Omega\to\Omega$2-invariant measure for the cocycle $\theta:\Omega\to\Omega$3 on the projective bundle. A measurable-reduction theorem, attributed in this context to Arnold–Cong–Oseledets or Barrientos–Malicet, then implies that, up to cohomology, such invariant measures are product measures

$\theta:\Omega\to\Omega$4

with $\theta:\Omega\to\Omega$5 a probability measure on $\theta:\Omega\to\Omega$6. The proof of generic nonvanishing of extremal exponents uses this product-measure rigidity together with perturbations that destroy any common invariant projective measure (Barrientos et al., 2018).

For volume-preserving piecewise $\theta:\Omega\to\Omega$7 random systems, a related invariance principle asserts that if

$\theta:\Omega\to\Omega$8

then there exists a probability measure $\theta:\Omega\to\Omega$9 on the projective bundle

$i$0

whose projection to $i$1 is $i$2, and which is invariant under the projectivized derivative: for $i$3-almost every $i$4,

$i$5

Equivalently, if $i$6 is a disintegration, then for $i$7-almost every $i$8,

$i$9

This places zero exponents in a rigid geometric regime (Magno et al., 13 Jan 2026).

Two applications make this rigidity explicit in dimension $C^r$0. For random additive perturbations of a billiard map associated with a strictly convex planar table on a surface of constant curvature, the Lyapunov exponents vanish almost everywhere if and only if the billiard table is a geodesic disk. For random additive perturbations of a standard map, the Lyapunov exponents vanish almost everywhere if and only if the standard map is integrable, equivalently $C^r$1. In the standard-map case, if $C^r$2, then $C^r$3 takes both hyperbolic and elliptic values on a set of positive measure, and a Furstenberg-type criterion in dimension $C^r$4 rules out a common invariant projective measure, so $C^r$5 (Magno et al., 13 Jan 2026).

These results support a general principle stated explicitly for conservative IID random maps: in the volume-preserving setting, vanishing Lyapunov exponents are extremely rigid, equivalent to the existence of a non-random invariant family of projective measures. In dimension $C^r$6, the dichotomy is especially transparent: positivity of $C^r$7 is equivalent to the absence of an invariant line field, while $C^r$8 signals full integrability (Magno et al., 13 Jan 2026).

4. Coexpansion on average, generator operators, and spectral gaps

A different line of work studies conservative IID random systems through the annealed transfer operator. Let $C^r$9 be a closed smooth Riemannian manifold with volume form $d\ge 2$00, and let $d\ge 2$01 be a probability measure on $d\ge 2$02 with compact support. The associated annealed transfer or generator operator acts on observables by

$d\ge 2$03

It is a Markov operator on $d\ge 2$04 with spectral radius $d\ge 2$05, and mass-conserving in the sense that $d\ge 2$06 (DeWitt et al., 10 Mar 2025).

The key hypothesis is that $d\ge 2$07 is coexpanding on average, also described as expanding on average on codimension-$d\ge 2$08 planes. This means that there exist $d\ge 2$09 and $d\ge 2$10 such that for every $d\ge 2$11 and every unit cotangent vector $d\ge 2$12,

$d\ge 2$13

Equivalently,

$d\ge 2$14

This condition is described as a weak hyperbolicity: instead of growth of $d\ge 2$15 along each orbit, only averaged logarithmic growth on codimension-$d\ge 2$16 planes is required (DeWitt et al., 10 Mar 2025).

On Sobolev spaces $d\ge 2$17 for sufficiently small $d\ge 2$18, the generator extends to a bounded operator with essential spectral radius strictly smaller than $d\ge 2$19: $d\ge 2$20 The proof uses symbol calculus, a Lasota–Yorke-type inequality, and a key estimate of the form

$d\ge 2$21

derived from the coexpansion hypothesis by a Taylor expansion in $d\ge 2$22. This quasi-compactness yields finitely many ergodic components for the stationary volume: there exist $d\ge 2$23 and a finite measurable partition $d\ge 2$24 so that $d\ge 2$25 is totally ergodic on each $d\ge 2$26 (DeWitt et al., 10 Mar 2025).

If, in addition, $d\ge 2$27 is totally ergodic, then $d\ge 2$28 on $d\ge 2$29 has an actual spectral gap. One then obtains exponential decay of correlations for zero-mean Hölder observables,

$d\ge 2$30

multiple exponential mixing for $d\ge 2$31 observables, and a central limit theorem for $d\ge 2$32 observables with variance given by the Green–Kubo formula

$d\ge 2$33

The same results include a Berry–Esseen bound $d\ge 2$34 for $d\ge 2$35 observables, and, if $d\ge 2$36 also coexpands on average, an extension of the CLT to $d\ge 2$37 observables. Small $d\ge 2$38 perturbations of $d\ge 2$39, including mildly dissipative ones with $d\ge 2$40 small, retain the essential spectral gap, so exponential mixing and the CLT are open properties in $d\ge 2$41 (DeWitt et al., 10 Mar 2025).

5. Annealed and quenched statistical properties

The annealed framework averages over the random driving, while the quenched framework studies almost every realization $d\ge 2$42. For a conservative IID random system $d\ge 2$43, annealed exponential mixing of the two-point motion on $d\ge 2$44 means that there exist constants $d\ge 2$45, $d\ge 2$46 such that for every pair of zero-mean observables $d\ge 2$47,

$d\ge 2$48

Quenched exponential mixing of the one-point motion on $d\ge 2$49 means that there exist $d\ge 2$50 and an almost surely finite random constant $d\ge 2$51 such that for every $d\ge 2$52 and every $d\ge 2$53,

$d\ge 2$54

The corresponding annealed and quenched CLTs are formulated in terms of ergodic sums $d\ge 2$55 (DeWitt et al., 5 Sep 2025).

A principal equivalence theorem states that annealed exponential mixing of the two-point motion implies quenched exponential mixing of the one-point motion. More precisely, if the two-point motion is annealed exponentially mixing on $d\ge 2$56, then for every $d\ge 2$57 there is $d\ge 2$58 so that for almost every $d\ge 2$59 there exists $d\ge 2$60, with

$d\ge 2$61

such that for all $d\ge 2$62 and all $d\ge 2$63,

$d\ge 2$64

By Sobolev–Hölder embedding and interpolation, the same statement holds on $d\ge 2$65 for every $d\ge 2$66 (DeWitt et al., 5 Sep 2025).

A second theorem adds an annealed CLT hypothesis for the two-point motion, with polynomial rate of convergence of characteristic functions, and deduces a quenched CLT for the one-point motion. Under compact support of $d\ge 2$67 on $d\ge 2$68, annealed two-point exponential mixing together with the annealed CLT implies that for every $d\ge 2$69, and likewise for every Hölder exponent $d\ge 2$70, almost every realization $d\ge 2$71 satisfies a quenched CLT for every zero-mean observable $d\ge 2$72 or $d\ge 2$73 (DeWitt et al., 5 Sep 2025).

The same work records two instructive failures of naive equivalence. IID uniform translations on $d\ge 2$74 mix perfectly in the annealed sense, since one iteration already makes $d\ge 2$75 uniform, but they have no quenched mixing. Conversely, a “$d\ge 2$76-type” example built from a random walk in $d\ge 2$77 acting by powers of a fixed Anosov map can have quenched mixing and quenched CLT but only polynomial decay of annealed correlations. This shows that the relationship between annealed and quenched behavior is subtle, and that the two-point motion is the relevant annealed object in the conservative IID setting (DeWitt et al., 5 Sep 2025).

6. Model classes, applications, and open directions

Several model classes realize the abstract hypotheses above. For coexpansion on average, one family consists of random toral automorphisms: if $d\ge 2$78 generate a Zariski-dense subgroup of $d\ge 2$79, and $d\ge 2$80, then the uniform measure on $d\ge 2$81 is coexpanding on average and totally ergodic, hence exponential mixing and the CLT hold; small volume-preserving $d\ge 2$82 perturbations remain strongly chaotic. A second family consists of random homogeneous actions on $d\ge 2$83, where $d\ge 2$84 is semisimple, $d\ge 2$85 is cocompact, and the generators lie in a Zariski-dense semisimple subgroup. A third family consists of generic $d\ge 2$86-tuples of isometries, or their small volume-preserving perturbations, on $d\ge 2$87 or more generally on an isotropic compact symmetric space; the system is either simultaneously conjugate back to isometries or coexpanding on average. Further examples include randomly switched flows generated by generic tuples of volume-preserving vector fields and perturbations of the Chirikov–Taylor standard map $d\ge 2$88 with random $d\ge 2$89, which is coexpanding on average for large $d\ge 2$90 (DeWitt et al., 10 Mar 2025).

The zero-exponent rigidity theory supplies complementary examples. Random additive perturbations of billiard maps on strictly convex $d\ge 2$91-smooth tables on the sphere, Euclidean plane, or hyperbolic plane fit the piecewise $d\ge 2$92 conservative framework; here vanishing Lyapunov exponents characterize geodesic disks. Random additive perturbations of standard maps on $d\ge 2$93 provide an explicit integrability criterion: $d\ge 2$94 almost everywhere if and only if $d\ge 2$95 (Magno et al., 13 Jan 2026).

Several open directions are identified explicitly. For generic nonvanishing of extremal exponents, the proof yields $d\ge 2$96-openness and $d\ge 2$97-density, but density is not known to be $d\ge 2$98-dense in full generality. Extensions to non-independent random compositions, such as Markov or stationary-ergodic systems, and to partially hyperbolic cocycles with lower regularity remain open. Understanding the finer structure of intermediate Lyapunov spectra under IID randomness is also described as a subject for future work (Barrientos et al., 2018).

Taken together, these results indicate that conservative IID random dynamical systems occupy a distinctive position between deterministic conservative dynamics and general random cocycle theory. In one direction, IID randomness can force generic non-uniform hyperbolicity of the extremal directions. In another, vanishing exponents imply strong projective-bundle rigidity and, in concrete two-dimensional models, characterize maximally symmetric or integrable cases. In the statistical direction, weak hyperbolicity in the form of coexpansion on average can produce quasi-compact annealed transfer operators, exponential mixing, and central limit theorems, while annealed control of the two-point motion can be transferred to quenched laws for almost every realization (Barrientos et al., 2018, DeWitt et al., 10 Mar 2025, DeWitt et al., 5 Sep 2025, Magno et al., 13 Jan 2026).