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On positive Lyapunov exponents and SRB measures for partially hyperbolic systems

This paper establishes that for $C^1$ diffeomorphisms with a dominated splitting, positive Lyapunov exponents along the center-unstable bundle imply non-uniform expansion and the existence of SRB measures, provided the bundle admits a finest 1-dominated splitting on the support of an observable measure.

Original authors: Reza Mohammadpour

Published 2026-07-13

📖 6 min read🧠 Deep dive

CC0 1.0

Original authors: Reza Mohammadpour

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

=== DRAFT ===
Imagine a giant, invisible dance floor (a compact manifold) where a magical DJ (a diffeomorphism) spins the music and shuffles the dancers around. Some dancers move in tight, predictable circles, while others zoom off in wild, chaotic directions. Mathematicians have long been trying to figure out: Which dancers are the "real" stars? These are the ones whose paths cover the most ground, the ones you'd actually see if you walked into the room and looked around. In the language of the paper, these are called physical measures.

But here's the tricky part: sometimes, the "real" stars are hiding. There might be a few dancers who look like they are everywhere, but they are actually just illusions. The paper introduces a new way to spot the true stars, called observable measures. Think of these as the "most likely" dancers. Even if a dancer isn't the absolute most popular (physical), they are still "observable" if you have a decent chance of bumping into them if you wander around the dance floor. The authors prove that these observable measures always exist, even when the true physical stars might be missing entirely.

The Main Discovery: The "Fastest" Speed Limit

The paper's biggest breakthrough is about measuring how fast these dancers are spinning in their chaotic directions (the center-unstable direction, or $E_{cu}$ ).

Usually, to know if a system is expanding chaotically, you need to check the speed of every single dancer. But the authors found a shortcut. They proved that if you look at the fastest spinning dancer among the "observable" group (the one with the highest average speed), that speed tells you everything you need to know about the whole room.

Here's the magic trick:

The Shortcut: You don't need to check every single point on the dance floor individually to find the maximum speed. You just need to find the "best" observable measure (the one with the highest average speed in the chaotic direction).
The Connection: If this "best" observable measure shows that the dancers are spinning fast (positive Lyapunov exponents), then almost every single dancer on the floor is also spinning fast.
The Result: This proves that the system is "non-uniformly expanding." In plain English: if the observable stars are moving fast, the whole dance floor is chaotic and expanding.

What the Paper Rules Out (The "No" List)

The authors are very careful about what they don't claim. They explicitly reject the idea that you need to check the entire dance floor to find the expansion in the sense of verifying every single point. Instead, they show that a condition holding for Lebesgue almost every x (which covers almost the entire floor) is sufficient. It's like saying you don't need to interview every single person in a city to know the average temperature; if the condition holds for the vast majority of the population (almost every point), the math works out.

Furthermore, they clarify that just because a measure is "observable" doesn't mean it's "physical" (the absolute most popular). There are cases where you have observable measures but no physical measures at all. The paper uses a specific example (Bowen's example) where a dancer becomes "indifferent" (stops moving fast enough to be a physical star) but remains observable.

Crucially, there is a limit to how rough the dance floor can be. While the paper works with slightly rough systems ( $C^1$ ) to prove the connection between observable measures and expansion, the final result guaranteeing the existence of SRB measures (the "gold standard" maps) requires the system to be slightly smoother ( $C^{1+\alpha}$ ). You cannot get the SRB measures for the roughest ( $C^1$ ) systems; the extra smoothness is a necessary condition for that specific treasure hunt.

How Sure Are They?

The authors aren't just guessing or running simulations; they have proved these facts.

Theorem A: They proved that the maximum speed you can find by looking at observable measures is exactly the same as the maximum speed you see if you look at almost every single point on the floor. This is a hard mathematical proof.
Theorem C: They proved that if the "fastest" observable dancer is spinning fast (meaning the smallest exponent is positive, which implies the maximizing measure is positive), then the system is definitely expanding. However, this only works if the dancers are holding their flashlights in a specific way. The paper requires a condition called a "1-dominated splitting" on the support of the observable measure. This means the chaotic directions must be "locked" into a specific geometric structure (a cone) for the math to hold. Without this "cone" condition, the proof that expansion leads to SRB measures fails.

The "SRB" Treasure Hunt

The ultimate goal of the paper is to find SRB measures. Think of an SRB measure as the "gold standard" map of the dance floor. It tells you exactly where the dancers will end up after a long time.

The paper shows that if you have a system with a specific kind of structure (a "dominated splitting," which is like having a clear separation between the calm dancers and the wild ones), and if the wild ones are spinning fast, and if the system is slightly smoother ( $C^{1+\alpha}$ ), then SRB measures definitely exist.

They also show that under these conditions, there are only a finite number of these SRB maps, and together, they cover almost the entire dance floor. This means that if you pick a random dancer, they will almost certainly end up following one of these SRB maps.

A Final Metaphor: The Cone of Vision

To make this work, the authors rely on a concept called a "1-dominated splitting." Imagine the dancers are holding flashlights. A "1-dominated splitting" means that for the wild dancers, there is a specific, narrow cone of light they are forced to shine in. They can't just point their flashlights anywhere; they are locked into a specific direction.

The paper proves that if these flashlights are locked into a cone on the "observable" dancers, then the math works out perfectly. You can trust the speed of the observable dancers to tell you the speed of the whole crowd. Without this "cone" condition, the math might break, but the authors show that as long as this cone exists on the observable group, the chaos is real and predictable in a statistical sense.

In short: The paper gives us a reliable way to spot the true chaos in a system by looking only at the most "observable" parts of it, proving that if the visible stars are spinning, the whole universe is spinning with them—provided the system is smooth enough and the dancers are moving in the right geometric formation.

Technical Summary: Positive Lyapunov Exponents and SRB Measures for Partially Hyperbolic Systems

Problem Statement
The paper addresses the relationship between Lyapunov exponents and the existence of Sinai–Ruelle–Bowen (SRB) measures in the context of partially hyperbolic dynamical systems. Specifically, it investigates $C^1$ diffeomorphisms on compact Riemannian manifolds admitting a dominated splitting $TM = E_{cs} \oplus E_{cu}$ . A central question in the field, originally posed by M. Viana, is whether the positivity of Lyapunov exponents on a set of positive Lebesgue measure implies the existence of SRB measures.

Previous work by Alves, Bonatti, and Viana established that if a system is non-uniformly expanding along the center-unstable bundle $E_{cu}$ (satisfying a specific limsup condition on the inverse derivative), then SRB measures exist. However, it remained an open question whether the weaker condition of positive Lyapunov exponents (positivity of the limit of the log-norm of the inverse derivative) implies this non-uniform expansion. Furthermore, the construction of SRB measures in these settings often required strong assumptions, such as uniform domination over the entire attractor or higher regularity ( $C^{1+\alpha}$ ).

Methodology
The author employs a combination of ergodic optimization theory, the theory of observable measures, and the geometric properties of dominated splittings.

Observable Measures: The paper utilizes the concept of observable measures (or physical-like measures), introduced by Catsigeras and Enrich. These are invariant measures whose basins of $\epsilon$ -attraction have positive Lebesgue measure. Unlike physical measures, observable measures always exist and form a compact set in the weak* topology.
Ergodic Optimization of Lyapunov Exponents: The author defines a maximizing observable measure as one that maximizes the Lyapunov exponent functional over the set of all observable measures. The paper proves that the essential supremum of the pointwise Lyapunov exponents (with respect to Lebesgue measure) coincides with the maximum Lyapunov exponent achieved by an observable measure.
Dominated Splittings and Cone Fields: A crucial technical assumption is that the cocycle $Df^{-1}|_{E_{cu}}$ admits a 1-dominated splitting on the support of an ergodic maximizing observable measure. This condition is weaker than requiring domination over the entire manifold or attractor. The existence of this splitting allows for the construction of invariant cone fields, which in turn ensures the super-multiplicativity of the norm of the cocycle iterates and the continuity of Lyapunov exponents with respect to the measure.
Tian's Theorem: The proof of the main result relies on a theorem by Tian regarding sequences of continuous functions satisfying sub-additivity and super-additivity conditions, linking the negativity of Lyapunov exponents for all observable measures to non-uniform expansion for Lebesgue almost every point.

Key Contributions and Results

Theorem A (Lyapunov Ergodic Optimization): The paper establishes that for a $C^1$ diffeomorphism with a dominated splitting $E \oplus F$ , if the cocycle $Df^{-1}|_F$ has a 1-dominated splitting on the support of an ergodic maximizing observable measure, then the essential supremum of the pointwise Lyapunov exponents equals the maximum Lyapunov exponent over all observable measures. This bridges the gap between pointwise behavior (Lebesgue typical) and measure-theoretic optimization.
Theorem C (Implication of Non-Uniform Expansion): The main result demonstrates that if the Lyapunov exponents along $E_{cu}$ are positive Lebesgue almost everywhere (equivalently, the limsup of the log-norm of the inverse derivative is negative), and the 1-dominated splitting assumption holds on the support of a maximizing observable measure, then the system is non-uniformly expanding along $E_{cu}$ . Specifically, there exist $\lambda > 0$ and $K \in \mathbb{N}$ such that the time-averaged contraction of the inverse derivative is bounded by $-\lambda$ for almost every point.
Existence of SRB Measures: As a consequence of Theorem C, the paper proves that for $C^{1+\alpha}$ diffeomorphisms with a dominated splitting $E_s \oplus E_{cu}$ (where $E_s$ is uniformly contracting), if the Lyapunov exponents along $E_{cu}$ are positive almost everywhere and the 1-dominated splitting condition holds, then the system admits finitely many ergodic physical SRB measures, and the union of their basins has full Lebesgue measure.
Corollary D: The paper further refines the relationship between physical, SRB, and observable measures, showing that under the stated conditions, the set of observable measures coincides with the set of physical SRB measures.

Significance and Claims
The paper claims to extend previous results on the existence of SRB measures by relaxing the assumptions required for the domination property. While earlier works often required domination over the entire attractor or uniform expansion properties, this work requires the 1-dominated splitting condition only on the support of an ergodic maximizing observable measure. Since such measures always exist, this is a significant weakening of the hypotheses.

The author notes that the results apply to partially hyperbolic attractors and provide a partial answer to the question of whether positive Lyapunov exponents imply non-uniform expansion. The paper explicitly states that the implication holds under the mild assumption of 1-domination on the support of a specific class of measures, rather than globally. The results also clarify the structure of the set of observable measures, showing that in the presence of these conditions, they are finite and coincide with physical SRB measures.

The paper does not claim to solve the general case for all $C^1$ diffeomorphisms without the domination assumption, nor does it propose new experimental applications. The focus remains strictly on the theoretical derivation of the existence of SRB measures from the positivity of Lyapunov exponents within the framework of dominated splittings and observable measures.