Mostly Contracting in Dynamical Systems

• Mostly contracting is a nonuniform hyperbolicity condition where the center or center-stable direction exhibits negative Lyapunov exponents on Lebesgue-positive subsets of unstable disks.
• It guarantees structured dynamics by establishing stable manifolds, Gibbs–Markov–Young towers, and strong statistical laws such as exponential decay of correlations and central limit theorems.
• The concept extends to random and fast-slow systems, providing a framework for classifying physical measures and maximal u-entropy states via geometric skeletons and inducing schemes.

Mostly contracting is a nonuniform hyperbolicity condition in which a center, center-stable, or analogous intermediate direction has negative Lyapunov behavior on the invariant measures or typical points relevant to the system. In partially hyperbolic dynamics, the condition appears in pointwise formulations on unstable disks and in measure-theoretic formulations on Gibbs $u$-states or SRB measures; in random dynamics, it is expressed by negativity of a maximal Lyapunov exponent defined through local Lipschitz constants. Across these settings, the condition supports a substantial structure theory: stable manifolds and absolute continuity, Gibbs–Markov–Young towers, skeletons of hyperbolic periodic points, finiteness and classification of physical measures, and strong statistical laws including exponential decay of correlations, central limit theorems, large deviations, and almost sure invariance principles (Alves et al., 27 Mar 2025, Mi et al., 2020, Barrientos et al., 2024).

1. Core definitions and equivalent criteria

For a $C^2$-diffeomorphism $f\colon M\to M$ with a forward-invariant compact attractor

$A=\bigcap_{n\ge 0} f^n(U),$

and a $Df$-invariant dominated splitting

$T_A M=E^{uu}\oplus E^{cs},$

the assumptions used in the Young-structure theory are uniform expansion on $E^{uu}$,

$\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$

and domination,

$$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$

for some metric and some $0<\lambda<1$. The largest central-stable Lyapunov exponent is

$C^2$0

In this formulation, “mostly contracting” means that on every local strong-unstable disk $C^2$1 there is a positive-Lebesgue-measure set of points $C^2$2 for which $C^2$3. An equivalent requirement is

$C^2$4

for a suitable $C^2$5-invariant SRB measure $C^2$6, but the pointwise condition on unstable disks is the standing hypothesis in that setting (Alves et al., 27 Mar 2025).

For partially hyperbolic diffeomorphisms with splitting

$C^2$7

the center bundle $C^2$8 is mostly contracting if for every Gibbs $C^2$9-state $f\colon M\to M$0,

$f\colon M\to M$1

or equivalently,

$f\colon M\to M$2

A closely related formulation requires that for any unstable disk $f\colon M\to M$3,

$f\colon M\to M$4

The same literature also uses

$f\colon M\to M$5

as an equivalent criterion in the Bonatti–Viana sense (Mi et al., 2020, Dolgopyat et al., 2014).

The term also has a broader random-dynamical meaning. For an i.i.d. random map $f\colon M\to M$6 generated by Lipschitz transformations on a compact metric space $f\colon M\to M$7, each stationary measure $f\colon M\to M$8 has a maximal Lyapunov exponent

$f\colon M\to M$9

and

$A=\bigcap_{n\ge 0} f^n(U),$0

The random map is mostly contracting if $A=\bigcap_{n\ge 0} f^n(U),$1. This is equivalent to each ergodic stationary measure having negative exponent, to a pointwise negative limsup for every $A=\bigcap_{n\ge 0} f^n(U),$2, and to the existence of $A=\bigcap_{n\ge 0} f^n(U),$3 and $A=\bigcap_{n\ge 0} f^n(U),$4 such that

$A=\bigcap_{n\ge 0} f^n(U),$5

(Barrientos et al., 2024).

A specialized fast-slow formulation appears for

$A=\bigcap_{n\ge 0} f^n(U),$6

on $A=\bigcap_{n\ge 0} f^n(U),$7, where the center is one-dimensional. There,

$A=\bigcap_{n\ge 0} f^n(U),$8

and the system is called mostly contracting if the spatial average of the induced observable $A=\bigcap_{n\ge 0} f^n(U),$9 at each attracting fixed point of the averaged dynamics is negative, concretely

$Df$0

After normalization, the standing assumption is

$Df$1

(Simoi et al., 2014).

2. Young structures, inducing schemes, and statistical laws

Under the attractor, domination, and pointwise mostly contracting hypotheses

$Df$2

one obtains a full Gibbs–Markov–Young tower. More precisely, there exist

$Df$3

continuous families of $Df$4-stable disks $Df$5, $Df$6-unstable disks $Df$7, a product set

$Df$8

and a countable partition $Df$9 into $T_A M=E^{uu}\oplus E^{cs},$0-subsets such that

$T_A M=E^{uu}\oplus E^{cs},$1

is an induced Gibbs–Markov map satisfying the usual properties $T_A M=E^{uu}\oplus E^{cs},$2–$T_A M=E^{uu}\oplus E^{cs},$3: Markov, uniform contraction on stable leaves, uniform backward contraction on unstable leaves, bounded distortion, and absolute continuity of holonomy. The return-time function $T_A M=E^{uu}\oplus E^{cs},$4 has exponential tails: $T_A M=E^{uu}\oplus E^{cs},$5 (Alves et al., 27 Mar 2025).

The construction proceeds through several geometric-combinatorial stages. First, Lebesgue-typical points on each strong-unstable disk have infinitely many hyperbolic times $T_A M=E^{uu}\oplus E^{cs},$6 such that

$T_A M=E^{uu}\oplus E^{cs},$7

for some $T_A M=E^{uu}\oplus E^{cs},$8, producing hyperbolic pre-balls by Pesin-type arguments. Second, adapting Chernov’s scheme, each local strong-unstable disk is refined into a Cantor-like partition $T_A M=E^{uu}\oplus E^{cs},$9 with “large” images, and the boundary term

$E^{uu}$0

is shown to decay exponentially in $E^{uu}$1. Third, one embeds unstable pieces into product-rectangles

$E^{uu}$2

and runs a capture–release–growth algorithm to obtain an auxiliary inducing scheme with exponentially small return tails. Finally, re-inducing on a single base rectangle $E^{uu}$3 ensures that every image $E^{uu}$4 $E^{uu}$5-crosses $E^{uu}$6, giving a full Young tower with exponential tails. Domination and bounded Jacobian distortion on unstable disks supply the required distortion estimates at each stage (Alves et al., 27 Mar 2025).

Once a Young tower with exponential return tails is available, standard tower machinery yields the usual statistical consequences. The cited results include existence, finiteness, and uniqueness statements for SRB measures, exponential decay of correlations,

$E^{uu}$7

for Hölder observables, the Central Limit Theorem,

$E^{uu}$8

exponential large deviations,

$E^{uu}$9

and a vector-valued almost sure invariance principle: for finite-dimensional Hölder $\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$0 not contained in a proper closed subspace,

$\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$1

for some $\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$2-dimensional Brownian motion $\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$3 and some $\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$4 (Alves et al., 27 Mar 2025).

3. Skeletons and the organization of physical measures

A central structural object for mostly contracting center dynamics is the skeleton. For a $\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$5 partially hyperbolic diffeomorphism with mostly contracting center, a skeleton is a finite collection of hyperbolic saddles

$\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$6

of maximum index, meaning $\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$7, satisfying two geometric conditions: full-unstable coverage, namely every unstable leaf meets

$\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$8

transversely, and absence of heteroclinic ties,

$\|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$9

The skeleton determines the physical measures: the assignment $$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$0 is a bijection between the skeleton and the physical measures, and

$$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$1

equals the homoclinic class of $$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$2, while

$$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$3

In particular, the system has exactly $$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$4 physical measures, their supports are pairwise disjoint, and their basins fill Lebesgue-almost every point of $$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$5 (Dolgopyat et al., 2014).

This picture is refined in the open class $$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$6, consisting of $$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$7 partially hyperbolic diffeomorphisms

$$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$8

for which the “upper center” bundle $$\|Df|_{E^{cs}(x)}\|\cdot \|Df^{-1}|_{E^{uu}(x)}\|<\lambda,$$9 is mostly expanding and the “lower center” bundle $0<\lambda<1$0 is mostly contracting. In this class, there are only finitely many ergodic physical measures $0<\lambda<1$1, the union of their basins has full Lebesgue measure, and these measures coincide with the extreme points of the convex, compact set of Gibbs $0<\lambda<1$2-states. A skeleton is defined as a finite set

$0<\lambda<1$3

of hyperbolic periodic points of stable index $0<\lambda<1$4 satisfying: every $0<\lambda<1$5-disk transverse to $0<\lambda<1$6 meets some $0<\lambda<1$7 transversely, and for $0<\lambda<1$8 there are no heteroclinic intersections between the stable manifold of $0<\lambda<1$9 and the unstable manifold of $C^2$00. Every $C^2$01 admits at least one skeleton; if $C^2$02 is $C^2$03, then the number of skeleton points equals the number of ergodic physical measures, and there is a bijection such that

$C^2$04

where $C^2$05 denotes the homoclinic class (Mi et al., 2020).

The proofs rely on Pesin-type blocks, Pliss arguments, abundance of hyperbolic periodic points via a Liao–Gan shadowing lemma, and a pruning procedure that removes heteroclinically related periodic orbits until a maximal family without such intersections remains. The resulting object is explicitly geometric and combinatorial: it records how unstable leaves are captured by stable manifolds of selected periodic points, and it encodes the number, supports, and basins of physical measures (Mi et al., 2020, Dolgopyat et al., 2014).

4. Perturbations, bifurcations, and stability

The skeleton formalism also governs variation under perturbation. If

$C^2$06

is a skeleton of $C^2$07, then under any $C^2$08-small perturbation $C^2$09, the continuations $C^2$10 form a pre-skeleton of $C^2$11. The number of physical measures of $C^2$12 is at most $C^2$13, and equality holds if and only if no new heteroclinic intersections appear among the $C^2$14. In the equality case, the physical measures $C^2$15 vary continuously in the weak$C^2$16 topology. Consequently, on $C^2$17 the number of ergodic physical measures is upper-semicontinuous in the $C^2$18-topology, and on each $C^2$19-open set where this number is constant, both the measures and their supports vary continuously in the weak$C^2$20 and Hausdorff topologies, respectively (Mi et al., 2020).

A complementary description emphasizes bifurcation by heteroclinic creation. Skeleton points persist to nearby systems, but new heteroclinic intersections among their continuations may appear. The number of physical measures never exceeds the size of the original skeleton and drops by one each time a heteroclinic tie among two formerly unconnected skeleton points is created. At such parameters, the corresponding basins become inseparable and one physical measure disappears: its basin is absorbed by the neighboring one. On open subsets where the same combinatorial skeleton persists, the basins vary continuously in the pseudo-distance

$C^2$21

The same framework produces examples with any given number of physical measures, with basins densely intermingled, and perturbations that can kill any chosen subset of the original measures or force a unique measure whose limit may be any convex combination of the original family (Dolgopyat et al., 2014).

Weak and strong forms of statistical stability coexist with this bifurcation picture. The entire class $C^2$22 is weakly statistically stable: if $C^2$23 in $C^2$24 and $C^2$25 is any sequence of physical measures for $C^2$26, then any limit point of $C^2$27 lies in the convex hull of the $C^2$28-physical measures. If $C^2$29 has a unique physical measure, then in a $C^2$30-neighborhood of $C^2$31 there is strong statistical stability, meaning continuity of the unique physical measure under perturbation (Mi et al., 2020).

There is also a stochastic analogue. For a $C^2$32 diffeomorphism admitting a dominated splitting

$C^2$33

with $C^2$34 mostly expanding and $C^2$35 mostly contracting, and with finitely many ergodic physical measures whose basins cover Lebesgue-almost every point of $C^2$36, every zero-noise limit of stationary measures under a regular random perturbation lies in the convex hull of the physical measures. This is the stochastic stability result proved via random dominated splittings, hyperbolic blocks, uniformly sized local random unstable manifolds, and absolute continuity of conditional measures on unstable stacks (Mi, 2020).

5. Maximal $C^2$37-entropy and support classes

In a $C^2$38 partially hyperbolic setting with an $C^2$39-Markov partition, the mostly contracting framework extends from physical measures to measures of maximal $C^2$40-entropy. The relevant class consists of maps admitting both a c-mostly contracting center along $C^2$41 and a c-mostly expanding center along $C^2$42. A c-Gibbs $C^2$43-state is a probability whose conditional measures on almost every strong-unstable plaque coincide with normalized leaf-volume. In this framework,

$C^2$44

so every maximal-$C^2$45 measure is a c-Gibbs $C^2$46-state. The ergodic maximal-$C^2$47 measures split into two types: c-$C^2$48-states and c-$C^2$49-states (Hangyue, 2023).

The classification theorem states that every ergodic maximal-$C^2$50 measure is one of these two types, but under the c-mostly expanding hypothesis no nontrivial c-$C^2$51-states of maximal $C^2$52-entropy can exist. Hence

$C^2$53

Under the c-mostly contracting hypothesis, the c-$C^2$54-states are described by a finite skeleton $C^2$55 of hyperbolic periodic points of stable index $C^2$56, with a bijection

$C^2$57

where $C^2$58 means $C^2$59. Under this correspondence,

$C^2$60

and every $C^2$61 arises in this way. Thus there are only finitely many support-classes of ergodic c-$C^2$62-states (Hangyue, 2023).

The variation theory is formulated in terms of support-classes rather than individual measures. For each $C^2$63 in the class, there is a $C^2$64-neighborhood $C^2$65 such that

$C^2$66

If the skeleton can be chosen to be a strong skeleton, then the map

$C^2$67

is upper-semicontinuous in $C^2$68, there is an open dense set on which it is locally constant, and on regions where the number of support-classes is fixed, the union of basins of representatives is dense in $C^2$69, the supports vary lower-semicontinuously in the Hausdorff metric, and each support is precisely the homoclinic class of its skeleton point. The paper also provides qualitative constructions in which any prescribed finite number $C^2$70 of distinct supports is realized, each support carries uncountably many ergodic measures, and small $C^2$71-perturbations reduce the number of support-classes from $C^2$72 to any $C^2$73 (Hangyue, 2023).

6. Random maps and fast-slow partially hyperbolic systems

For random Lipschitz maps on compact metric spaces, mostly contracting is defined without differentiability of the phase dynamics. The essential input is the local Lipschitz cocycle

$C^2$74

the stationarity condition for $C^2$75, and the negativity of

$C^2$76

This class is open in the topology induced by the maps and their Lipschitz constants. Under the local contraction property established in the paper, any continuous random map admits finitely many ergodic stationary measures $C^2$77, and the phase space decomposes as

$C^2$78

uniformly in $C^2$79 and for $C^2$80-almost every $C^2$81. The same work proves the strong law of large numbers for non-uniquely ergodic systems, the limit theorem for the law of random iterations, the global Palis conjecture for random maps, and quasi-compactness of the annealed Koopman operator

$C^2$82

With an exponential-moment condition

$C^2$83

the operator $C^2$84 on $C^2$85 has a spectral gap, implying a Central Limit Theorem for $C^2$86 observables, a Large-Deviations Principle with quadratic bound in $C^2$87, statistical stability, and continuity and Hölder continuity of Lyapunov exponents. The examples listed in the paper include projective actions of locally constant linear cocycles under simplicity of the first Lyapunov exponent and quasi-irreducibility, random circle $C^2$88 diffeomorphisms with no common invariant measure, interval $C^2$89 diffeomorphisms onto their images under a no-common-two-point-invariant-set assumption, and $C^2$90 diffeomorphisms of a Cantor set on a line with no common invariant measure (Barrientos et al., 2024).

A different specialized realization occurs in fast-slow partially hyperbolic systems on $C^2$91,

$C^2$92

with uniformly expanding $C^2$93-dynamics and one-dimensional center. Under hypotheses $C^2$94–$C^2$95 and the generic condition $C^2$96, for sufficiently small $C^2$97 the map $C^2$98 admits at most $C^2$99 SRB measures. If the completeness condition $f\colon M\to M$00 holds, then $f\colon M\to M$01 and there is a unique SRB measure $f\colon M\to M$02. Each SRB measure is supported in a small trapping set near one of the sinks $f\colon M\to M$03, and its ergodic basin contains a positive-Lebesgue-measure neighborhood around that sink. The same analysis gives sharp exponential decay of correlations: for suitable $f\colon M\to M$04 and $f\colon M\to M$05,

$f\colon M\to M$06

with

$f\colon M\to M$07

The proof combines averaging, large deviations, a local central limit theorem, standard pairs and families, center-cone contraction, holonomy and distortion control, and a recursive coupling argument (Simoi et al., 2014).

Taken together, these developments show that “mostly contracting” is not a single theorem but a family of closely related hypotheses adapted to several nonuniformly hyperbolic settings. In deterministic partially hyperbolicity it organizes the geometry of stable manifolds, inducing schemes, and skeletons; in maximal-$f\colon M\to M$08-entropy theory it yields finiteness of support-classes; in stochastic and random contexts it supports stationary-measure classification and zero-noise selection; and in fast-slow models it leads to explicit SRB classifications and nearly optimal decay-rate estimates (Alves et al., 27 Mar 2025, Hangyue, 2023, Barrientos et al., 2024).