Mostly Expanding Center in Partially Hyperbolic Dynamics

• Mostly expanding center is a measure-theoretic nonuniform hyperbolicity condition for partially hyperbolic diffeomorphisms that requires every Gibbs u-state to have positive center Lyapunov exponents.
• The approach leverages an integrated averaged expansion condition, enabling C¹-openness and robust statistical stability through entropy semicontinuity.
• Extensions of the concept influence physical measures, skeleton structures, and fast–slow systems, leading to strong mixing properties and precise entropy characterizations.

Mostly expanding center is a measure-theoretic nonuniform hyperbolicity condition for partially hyperbolic dynamics. In the standard setting one considers a partially hyperbolic diffeomorphism $f$ with invariant splitting

$TM = E^s \oplus E^c \oplus E^u,$

where $Df|_{E^s}$ is uniformly contracting, $Df|_{E^u}$ is uniformly expanding, and $Df|_{E^c}$ is intermediate. The modern formulation treats the center direction as “mostly expanding” not by a pointwise requirement of uniform expansion, but by demanding that every Gibbs $u$-state has positive center Lyapunov exponents almost everywhere. This stronger Gibbs-$u$-state formulation, emphasized in “Entropy along expanding foliations” (Yang, 2016), is the version used because it is the right notion for openness; it subsequently became the organizing hypothesis for results on physical measures, skeletons, statistical stability, entropy structure, stochastic stability, flows, and fast–slow partially hyperbolic systems (Yang, 2019, Andersson et al., 2017, Mi et al., 2020, Mi, 2020, Mi et al., 2020, Zhang, 2024, Simoi et al., 17 Nov 2025).

1. Definition and conceptual role

For $C^{1+\alpha}$ or $C^r$, $r>1$, partially hyperbolic diffeomorphisms, a Gibbs $TM = E^s \oplus E^c \oplus E^u,$0-state is an invariant measure whose conditional measures along strong unstable leaves are absolutely continuous with respect to leaf volume. In this framework, a diffeomorphism is mostly expanding along the central direction if every Gibbs $TM = E^s \oplus E^c \oplus E^u,$1-state has positive center Lyapunov exponents almost everywhere (Yang, 2016, Yang, 2019, Andersson et al., 2017).

This definition is explicitly stronger than older formulations in the literature. “Entropy along expanding foliations” (Yang, 2016) contrasts it with the original Alves–Bonatti–Viana notion and states that the stronger Gibbs-$TM = E^s \oplus E^c \oplus E^u,$2-state formulation is the one used because it is the right notion for openness. “Statistical stability of mostly expanding diffeomorphisms” (Andersson et al., 2017) makes the same point: the condition is stronger than the older NUE-condition and, unlike the NUE-condition, it is robust/open.

A useful integrated formulation appears in (Yang, 2016). Proposition 5.5 states that $TM = E^s \oplus E^c \oplus E^u,$3 has mostly expanding center if and only if there exist $TM = E^s \oplus E^c \oplus E^u,$4 and $TM = E^s \oplus E^c \oplus E^u,$5 such that for every $TM = E^s \oplus E^c \oplus E^u,$6,

$TM = E^s \oplus E^c \oplus E^u,$7

The same paper gives an improved form: if $TM = E^s \oplus E^c \oplus E^u,$8 has mostly expanding center, then for some $TM = E^s \oplus E^c \oplus E^u,$9 and $Df|_{E^s}$0,

$Df|_{E^s}$1

for every $Df|_{E^s}$2 (Yang, 2016). This identifies mostly expanding center as an averaged expansion condition on all Gibbs $Df|_{E^s}$3-states rather than a pointwise property of individual orbits.

The same idea extends beyond the basic $Df|_{E^s}$4 setting. In mixed-center frameworks one works with

$Df|_{E^s}$5

and requires that every Gibbs $Df|_{E^s}$6-state has only positive Lyapunov exponents along $Df|_{E^s}$7 and only negative Lyapunov exponents along $Df|_{E^s}$8 (Mi et al., 2020, Mi, 2020). This suggests that “mostly expanding center” is best understood as a sign condition on Lyapunov data for dynamically relevant invariant measures.

2. Entropy along expanding foliations and $Df|_{E^s}$9-openness

The decisive structural advance in (Yang, 2016) is the upper semicontinuity of entropy along expanding foliations. For diffeomorphisms $Df|_{E^u}$0 in $Df|_{E^u}$1, invariant measures $Df|_{E^u}$2 in the weak$Df|_{E^u}$3 topology, and expanding foliations $Df|_{E^u}$4, the partial entropy satisfies

$Df|_{E^u}$5

This upper semicontinuity is the mechanism behind the openness of the mostly expanding center condition (Yang, 2016).

The same paper uses the entropy characterization of Gibbs $Df|_{E^u}$6-states. For a $Df|_{E^u}$7 partially hyperbolic diffeomorphism,

$Df|_{E^u}$8

and equality holds if and only if $Df|_{E^u}$9 is a Gibbs $Df|_{E^c}$0-state: $Df|_{E^c}$1 If $Df|_{E^c}$2 are Gibbs $Df|_{E^c}$3-states for $Df|_{E^c}$4, then

$Df|_{E^c}$5

and the semicontinuity theorem yields

$Df|_{E^c}$6

Since the unstable Jacobians converge uniformly, one recovers the equality needed to conclude that the weak$Df|_{E^c}$7 limit is again a Gibbs $Df|_{E^c}$8-state. This is Theorem B of (Yang, 2016): the set-valued map $Df|_{E^c}$9 varies upper semicontinuously in the $u$0 topology among $u$1 partially hyperbolic diffeomorphisms.

The openness statement is Theorem C of (Yang, 2016): the sets of partially hyperbolic diffeomorphisms with mostly contracting center or mostly expanding center are $u$2 open, in the sense that every $u$3 partially hyperbolic diffeomorphism with mostly contracting or mostly expanding center admits a $u$4 open neighborhood such that every $u$5 diffeomorphism in this neighborhood has the same property. Later work repeatedly uses this openness as a starting point (Andersson et al., 2017, Yang, 2019, Mi et al., 2020).

A common misconception is to read “mostly expanding” as a uniform pointwise condition on $u$6. The foundational papers do not use that interpretation. They formulate the condition through all Gibbs $u$7-states and then deduce robustness from entropy semicontinuity and Gibbs-$u$8-state stability (Yang, 2016, Andersson et al., 2017).

3. Physical measures, basins, and the skeleton structure

A physical measure is an invariant probability measure whose basin has positive volume. In the mostly expanding setting, the central questions are how many such measures exist, how their supports are organized, and how their basins fill phase space (Yang, 2019, Andersson et al., 2017).

“Geometrical and measure-theoretic structures of maps with mostly expanding center” (Yang, 2019) introduces the central geometric object: a skeleton. For a partially hyperbolic $u$9 with splitting $u$0, an index $u$1 skeleton is a finite set

$u$2

of hyperbolic periodic saddles with stable index $u$3 such that the union of their stable manifolds is dense and the set is minimal with that property. A pre-skeleton satisfies only the density requirement (Yang, 2019).

The core theorem in (Yang, 2019) states that if $u$4 is $u$5, partially hyperbolic, and has mostly expanding center, then $u$6 admits an index $u$7 skeleton, and each skeleton element corresponds to a distinct physical measure. More precisely, if $u$8 is a skeleton, then the number of physical measures is exactly $u$9, and for each $C^{1+\alpha}$0,

$C^{1+\alpha}$1

while

$C^{1+\alpha}$2

The interiors of the basin closures are disjoint (Yang, 2019).

The measure-theoretic side of the same paper is expressed through

$C^{1+\alpha}$3

$C^{1+\alpha}$4

and

$C^{1+\alpha}$5

For $C^{1+\alpha}$6 maps with mostly expanding center, the extreme points of $C^{1+\alpha}$7 are exactly the physical measures, $C^{1+\alpha}$8 is compact and convex, and it has finitely many ergodic extreme points (Yang, 2019).

Related results in (Yang, 2016) already establish strong statistical consequences. For mostly expanding center systems, there exist finitely many physical measures, and in the robust neighborhood from Proposition 6.9 the basins of these measures cover full volume, each containing a ball of uniform size in Lebesgue-a.e. sense. In the accessible volume-preserving one-dimensional-center setting of Theorem D, nearby $C^{1+\alpha}$9 systems have a unique physical measure whose basin has full volume (Yang, 2016).

A later entropy-oriented development, (Hangyue, 2023), adapts skeleton methods to maximal $C^r$0-entropy measures in a setting with $C^r$1-Markov partitions and $C^r$2-mostly expanding center. There the relevant $C^r$3-$C^r$4-states admit a support-and-basin description analogous to physical measures, and the number of support classes is finite in the sense that the measure supports are consistent (Hangyue, 2023). This suggests a broader principle: once center expansion is imposed on all relevant Gibbs-like states, skeleton-type combinatorics tends to govern both SRB-type and maximal $C^r$5-entropy structures.

4. Statistical stability, bifurcation, and perturbations

The mostly expanding condition supports several layers of statistical stability. “Statistical stability of mostly expanding diffeomorphisms” (Andersson et al., 2017) proves that for the open class of $C^r$6, $C^r$7, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs $C^r$8-state are positive, physical measures vary continuously with the dynamics whenever possible. In the transitive case, there is a $C^r$9 neighborhood in which every nearby map has a unique physical measure, and the map

$r>1$0

is continuous in the weak$r>1$1 topology (Andersson et al., 2017).

In the non-transitive case, (Andersson et al., 2017) proves upper semicontinuity of the number of physical measures and continuity of the individual physical measures along parameter regions where that number remains constant. A weaker form of statistical stability is also established: any weak$r>1$2 limit of physical measures of nearby maps is a convex combination of physical measures of the limit map (Andersson et al., 2017).

A key technical innovation in (Andersson et al., 2017) is a Pliss-like lemma yielding hyperbolic times with frequency close to one. The lemma shows that if most terms in a sequence lie above a threshold $r>1$3 and all are bounded below by $r>1$4, then one obtains many Pliss times $r>1$5, with density that can be made arbitrarily close to $r>1$6. This is used to construct large Pesin blocks and uniform local $r>1$7-manifolds for nearby systems and nearby Gibbs $r>1$8-states (Andersson et al., 2017).

The skeleton perspective sharpens the perturbative picture. In (Yang, 2019), the number of physical measures is upper semi-continuous in a $r>1$9 neighborhood, locally constant on a $TM = E^s \oplus E^c \oplus E^u,$00-open and dense subset, and physical measures vary continuously in the weak$TM = E^s \oplus E^c \oplus E^u,$01 topology there. The bifurcation mechanism is explicitly geometric: if $TM = E^s \oplus E^c \oplus E^u,$02 is a skeleton of $TM = E^s \oplus E^c \oplus E^u,$03, then for nearby maps $TM = E^s \oplus E^c \oplus E^u,$04 the continuations $TM = E^s \oplus E^c \oplus E^u,$05 form a pre-skeleton; if no heteroclinic intersections appear among them, the number of physical measures is preserved, while the appearance of heteroclinic intersections can merge physical measures and decrease that number (Yang, 2019).

“Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers” (Mi et al., 2020) extends this philosophy to mixed-center classes $TM = E^s \oplus E^c \oplus E^u,$06. There a skeleton is defined by transverse intersection with any $TM = E^s \oplus E^c \oplus E^u,$07 disk transverse to $TM = E^s \oplus E^c \oplus E^u,$08, together with absence of heteroclinic intersections among distinct skeleton points. The paper proves a one-to-one correspondence between skeleton periodic points and physical measures, and shows weak statistical stability in the $TM = E^s \oplus E^c \oplus E^u,$09 topology; in the unique physical measure case, this yields ordinary continuity of the physical measure under perturbation (Mi et al., 2020).

5. Consequences for entropy, transitivity, and mixing

Mostly expanding center has consequences far beyond mere existence of SRB-type measures. In (Yang, 2016), Theorem F states that every $TM = E^s \oplus E^c \oplus E^u,$10 volume preserving, accessible partially hyperbolic diffeomorphism with one-dimensional center and non-vanishing center exponent is $TM = E^s \oplus E^c \oplus E^u,$11 robustly transitive. The proof again uses entropy inequalities and the robustness of the Gibbs-$TM = E^s \oplus E^c \oplus E^u,$12-state structure (Yang, 2016).

The same paper supplies a new class of mostly expanding examples. Theorem D states: if $TM = E^s \oplus E^c \oplus E^u,$13 is a $TM = E^s \oplus E^c \oplus E^u,$14 volume preserving partially hyperbolic diffeomorphism with one-dimensional center, the center exponent of the volume measure is positive, and $TM = E^s \oplus E^c \oplus E^u,$15 is accessible, then $TM = E^s \oplus E^c \oplus E^u,$16 admits a $TM = E^s \oplus E^c \oplus E^u,$17 open neighborhood, among diffeomorphisms not necessarily volume preserving, such that every $TM = E^s \oplus E^c \oplus E^u,$18 diffeomorphism in this neighborhood has mostly expanding center and admits a unique physical measure whose basin has full volume (Yang, 2016). Corollary E further gives a residual subset in that neighborhood for which every $TM = E^s \oplus E^c \oplus E^u,$19 diffeomorphism has a unique physical measure with full basin (Yang, 2016).

The entropy structure of mostly expanding systems is studied directly in “Entropy properties of mostly expanding partially hyperbolic diffeomorphisms” (Zhang, 2024). For partially hyperbolic $TM = E^s \oplus E^c \oplus E^u,$20 with $TM = E^s \oplus E^c \oplus E^u,$21, positive center exponent for all $TM = E^s \oplus E^c \oplus E^u,$22, and minimal strong stable foliation, there exists a $TM = E^s \oplus E^c \oplus E^u,$23-open neighborhood in which topological entropy varies continuously and the intermediate entropy property holds (Zhang, 2024). The technical engine is that every non-hyperbolic ergodic measure can be approximated by horseshoes both in entropy and in the weak$TM = E^s \oplus E^c \oplus E^u,$24 topology (Zhang, 2024).

That paper also studies the zero-center Lyapunov set

$TM = E^s \oplus E^c \oplus E^u,$25

showing that its entropy can be approximated by hyperbolic basic sets with uniformly expanding center and arbitrarily small positive center exponents (Zhang, 2024). A plausible implication is that mostly expanding center creates enough hyperbolic approximation to organize even the non-hyperbolic part of the ergodic decomposition.

Strong statistical properties are also available. In (Yang, 2019), physical measures for maps with mostly expanding center satisfy exponential decay of correlations for Hölder observables. More precisely, for each physical measure $TM = E^s \oplus E^c \oplus E^u,$26 of $TM = E^s \oplus E^c \oplus E^u,$27,

$TM = E^s \oplus E^c \oplus E^u,$28

for some $TM = E^s \oplus E^c \oplus E^u,$29; the paper also derives a central limit theorem, exponential large deviations, and the variance formula (Yang, 2019).

6. Generalizations and specialized settings

Several later works adapt the notion of mostly expanding center to settings that differ substantially from the original discrete-time, one-center-bundle picture.

For partially hyperbolic flows, “SRB measures for partially hyperbolic flows with mostly expanding center” (Mi et al., 2020) uses a 2D center bundle and a Gibbs sectional expanding condition. The center bundle $TM = E^s \oplus E^c \oplus E^u,$30 is called Gibbs sectional expanding if $TM = E^s \oplus E^c \oplus E^u,$31 and for every Gibbs $TM = E^s \oplus E^c \oplus E^u,$32-state $TM = E^s \oplus E^c \oplus E^u,$33,

$TM = E^s \oplus E^c \oplus E^u,$34

for every 2-dimensional subspace $TM = E^s \oplus E^c \oplus E^u,$35, for $TM = E^s \oplus E^c \oplus E^u,$36-a.e. $TM = E^s \oplus E^c \oplus E^u,$37 (Mi et al., 2020). Under $TM = E^s \oplus E^c \oplus E^u,$38, the attractor supports an SRB measure, and in the $TM = E^s \oplus E^c \oplus E^u,$39 case with Gibbs sectional expanding center there are finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin; under transitivity the measure is unique (Mi et al., 2020).

For random perturbations, “Stochastic stability for partially hyperbolic diffeomorphisms with mostly expanding and contracting centers” (Mi, 2020) considers an open set $TM = E^s \oplus E^c \oplus E^u,$40 of $TM = E^s \oplus E^c \oplus E^u,$41 diffeomorphisms with splitting

$TM = E^s \oplus E^c \oplus E^u,$42

where every Gibbs $TM = E^s \oplus E^c \oplus E^u,$43-state has only positive Lyapunov exponents along $TM = E^s \oplus E^c \oplus E^u,$44 and only negative Lyapunov exponents along $TM = E^s \oplus E^c \oplus E^u,$45. The main theorem states that every $TM = E^s \oplus E^c \oplus E^u,$46 is stochastically stable: every zero-noise limit of stationary measures lies in the convex hull of the physical measures of $TM = E^s \oplus E^c \oplus E^u,$47 (Mi, 2020). The proof passes through hyperbolic blocks and random Gibbs $TM = E^s \oplus E^c \oplus E^u,$48-states.

For fast–slow systems, “Statistical properties of mostly expanding fast-slow partially hyperbolic systems” (Simoi et al., 17 Nov 2025) studies $TM = E^s \oplus E^c \oplus E^u,$49 maps on $TM = E^s \oplus E^c \oplus E^u,$50 of the form

$TM = E^s \oplus E^c \oplus E^u,$51

with $TM = E^s \oplus E^c \oplus E^u,$52 after passing to an iterate (Simoi et al., 17 Nov 2025). There the decisive quantity is

$TM = E^s \oplus E^c \oplus E^u,$53

with fiber average

$TM = E^s \oplus E^c \oplus E^u,$54

The system is mostly expanding if

$TM = E^s \oplus E^c \oplus E^u,$55

at the sink $TM = E^s \oplus E^c \oplus E^u,$56 of the averaged slow dynamics (Simoi et al., 17 Nov 2025). Under the non-cohomology condition defining $TM = E^s \oplus E^c \oplus E^u,$57, the paper proves existence of a unique physical measure $TM = E^s \oplus E^c \oplus E^u,$58, absolute continuity $TM = E^s \oplus E^c \oplus E^u,$59, and exponential decay of correlations: $TM = E^s \oplus E^c \oplus E^u,$60 with

$TM = E^s \oplus E^c \oplus E^u,$61

The paper emphasizes a paradoxical feature: although $TM = E^s \oplus E^c \oplus E^u,$62 is a sink for the averaged dynamics, in the mostly expanding case the center Lyapunov exponent is positive for the physical measure, tied to the non-absolute continuity of the center foliation (Simoi et al., 17 Nov 2025).

These extensions indicate that the core content of “mostly expanding center” is not tied to a single model class. What persists across settings is the same principle: one imposes positivity of center or center-like exponents on all Gibbs $TM = E^s \oplus E^c \oplus E^u,$63-states or their analogues, and from that obtains robust statistical organization, finite or unique physical measures, and strong entropy or mixing properties.