Conditional Variational Principle

• Conditional Variational Principle is a framework that characterizes variational formulations after imposing auxiliary restrictions in diverse mathematical settings.
• It equates topological and measure-theoretic entropies under constraints such as empirical measures, historic behavior, and unstable dynamics.
• The principle extends to applications like reverse Noether theorems in PDEs and conditional Ekeland variational methods in random metric spaces.

Searching arXiv for recent and relevant papers on "Conditional Variational Principle" across dynamical systems, PDE inverse problems, skew products, partial hyperbolicity, and random metric spaces. Conditional variational principle is a name used in the cited literature for several mathematically distinct statements in which a variational characterization is available only after auxiliary restrictions are imposed. In topological dynamics, the restriction may be a condition on the set of empirical-measure limit points, on historic behavior, or on fiberwise level sets of Birkhoff averages (Yin et al., 2016, Yin et al., 2015, Liu et al., 2024). In the inverse problem of the calculus of variations, the restriction is a pair of hypotheses on symmetries and corresponding conservation laws for a second-order source form, from which local variationality is deduced (Dafinger, 2019). In random analysis, the restriction is encoded by almost-sure order, random topologies, and locality conditions for $\bar L^0$-valued functionals, yielding a conditional form of Ekeland’s variational principle (Guo et al., 2011). A related variational principle in partially hyperbolic dynamics isolates entropy generated by the unstable foliation and identifies unstable topological entropy with the supremum of unstable metric entropies (Hu et al., 2017).

1. Empirical measures, entropy of constrained sets, and shadowing-based conditional principles

For a topological dynamical system $(X,d,f)$, the basic empirical object is

$$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$

and $V(x)$ denotes the set of weak-* limit points of $\mathscr E_n(x)$. Given $I\subset M_{\mathrm{inv}}(X,f)$, the constrained sets are

$$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$

The entropy of such noncompact sets is taken in Bowen’s sense through covers by Bowen balls $B_n(x,\varepsilon)$ and the Carathéodory-type critical value $h_{\mathrm{top}}(f,Z)$ (Yin et al., 2016).

Under the hypotheses that $(X,f)$ is topologically transitive and has the pseudo-orbit tracing property, the principal conditional variational principles identify the entropy of these constrained sets with a supremum of measure-theoretic entropies. If $(X,d,f)$0 is a nonempty open subset, then

$(X,d,f)$1

If $(X,d,f)$2 is convex with $(X,d,f)$3, then the same equality holds with $(X,d,f)$4 replaced by $(X,d,f)$5. A shrinking principle is also proved: if $(X,d,f)$6 and the entropy map $(X,d,f)$7 is upper semicontinuous, then

$(X,d,f)$8

For nontransitive systems with POTP, analogous statements are localized to $(X,d,f)$9 for an ergodic measure $$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$0 (Yin et al., 2016).

The upper bound is obtained from Bowen’s lemma: if

$$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$1

then $$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$2. The lower bound uses Katok’s entropy formula, approximation of invariant measures by finite rational convex combinations of ergodic measures, concatenation of pseudo-orbits, transitivity for short transition pieces, and POTP to shadow the resulting pseudo-orbits by true orbits. This is the mechanism that makes the principle conditional: the entropy formula is realized only after prescribing admissible asymptotic measure behavior (Yin et al., 2016).

The same framework yields multifractal constraints for Birkhoff averages. For $$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$3 and $$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$4, the sets of points whose accumulation values of $$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$5 lie in, or intersect, $$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$6 satisfy analogous entropy formulas when $$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$7 is open or convex with nonempty interior. This connects conditional variational principles for $$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$8 directly to entropy spectra of Birkhoff averages (Yin et al., 2016).

2. Historic sets in nonuniformly hyperbolic dynamics

In the Pesin-theoretic setting, let $$\mathscr{E}_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^ix},$$9 be a $V(x)$0 diffeomorphism of a compact Riemannian manifold and let $V(x)$1 be a nonempty Pesin set. For a continuous observable $V(x)$2, the historic set is

$V(x)$3

and for an ergodic measure $V(x)$4 one defines

$V(x)$5

The conditional set $V(x)$6 restricts attention to points whose empirical measures have limit points supported on the Pesin-saturated set associated with $V(x)$7 (Yin et al., 2015).

The central theorem states a dichotomy. For $V(x)$8, either the map $V(x)$9 is constant on $\mathscr E_n(x)$0, or the historic set intersects $\mathscr E_n(x)$1 nontrivially and

$\mathscr E_n(x)$2

In particular, if $\mathscr E_n(x)$3 is not constant on $\mathscr E_n(x)$4, then $\mathscr E_n(x)$5 is non-empty and

$\mathscr E_n(x)$6

The result is explicitly an entropy conditional variational principle; the paper does not state pressure equalities $\mathscr E_n(x)$7 and works with $\mathscr E_n(x)$8 (Yin et al., 2015).

The proof again splits into upper and lower bounds, but the tools are adapted to nonuniform hyperbolicity. The upper bound uses Bowen’s entropy estimate for sets defined by entropy bounds on limit measures. The lower bound chooses two invariant measures $\mathscr E_n(x)$9 with distinct $I\subset M_{\mathrm{inv}}(X,f)$0-averages and entropy close to

$I\subset M_{\mathrm{inv}}(X,f)$1

approximates them by convex combinations of ergodic measures, extracts large separated sets from Pesin blocks using Katok’s entropy formula, and concatenates orbit segments by the weak shadowing lemma on Pesin sets. The resulting shadowing points alternate between measure profiles close to $I\subset M_{\mathrm{inv}}(X,f)$2 and $I\subset M_{\mathrm{inv}}(X,f)$3, so the Birkhoff averages of $I\subset M_{\mathrm{inv}}(X,f)$4 oscillate and do not converge (Yin et al., 2015).

Applications include $I\subset M_{\mathrm{inv}}(X,f)$5 surface diffeomorphisms with positive topological entropy, hyperbolic measures, Katok maps on $I\subset M_{\mathrm{inv}}(X,f)$6, and robustly transitive partially hyperbolic diffeomorphisms derived from Anosov systems. In these cases, the conditional principle shows that historic sets, although of zero measure for any invariant measure by Birkhoff’s theorem, can be topologically large in the precise sense of carrying entropy equal to the supremum of measure entropies over the relevant conditional class (Yin et al., 2015).

3. Fiberwise conditional variational principles for skew products

For a skew product

$I\subset M_{\mathrm{inv}}(X,f)$7

on $I\subset M_{\mathrm{inv}}(X,f)$8, the cited work assumes that $I\subset M_{\mathrm{inv}}(X,f)$9 is a uniquely ergodic homeomorphism with unique invariant Borel probability measure $$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$0, that the skew product is Anosov on fibers, and that it is topological mixing on fibers. The fiber Bowen metric is

$$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$1

and the associated Carathéodory construction defines the fiber Bowen’s topological entropy $$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$2 of a set $$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$3 along the fiber $$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$4 (Liu et al., 2024).

Given a continuous observable $$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$5 and $$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$6, the classical level set is

$$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$7

and the fiber slice is $$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$8. Let

$$\Delta_{\mathrm{sub}}(I)=\{x\in X:V(x)\subset I\},\qquad \Delta_{\mathrm{cap}}(I)=\{x\in X:V(x)\cap I\neq\emptyset\}.$$9

where $B_n(x,\varepsilon)$0. Theorem B states that for $B_n(x,\varepsilon)$1 and for $B_n(x,\varepsilon)$2-almost every $B_n(x,\varepsilon)$3,

$B_n(x,\varepsilon)$4

where

$B_n(x,\varepsilon)$5

This is the fiber-level conditional variational principle for Birkhoff averages (Liu et al., 2024).

The same paper proves that Anosov and topological mixing on fibers imply the fiber specification property, and moreover the associated random dynamical system has the specification property in the sense of Gundlach and Kifer. The proof of the variational principle uses this specification mechanism to carry out a fiber Moran-like construction, producing Cantor-like sets $B_n(x,\varepsilon)$6 and measures on them with uniform Bowen-ball estimates. The upper bound is expressed through an asymptotic fiber growth rate $B_n(x,\varepsilon)$7 derived from covering and separated numbers on deviation sets, while the lower bound comes from saturation under fiber specification (Liu et al., 2024).

A pressure formulation is also available. Writing $B_n(x,\varepsilon)$8 for the fiber topological pressure and $B_n(x,\varepsilon)$9, the Legendre transform

$h_{\mathrm{top}}(f,Z)$0

satisfies

$h_{\mathrm{top}}(f,Z)$1

for $h_{\mathrm{top}}(f,Z)$2 and

$h_{\mathrm{top}}(f,Z)$3

for $h_{\mathrm{top}}(f,Z)$4, for $h_{\mathrm{top}}(f,Z)$5-almost every $h_{\mathrm{top}}(f,Z)$6. Under the same hypotheses, every Hölder continuous potential has a unique equilibrium state $h_{\mathrm{top}}(f,Z)$7 with the Gibbs property, and an analogous conditional variational principle holds for fiber slices $h_{\mathrm{top}}(f,Z)$8 of local entropy level sets (Liu et al., 2024).

Examples include fiber Anosov maps on $h_{\mathrm{top}}(f,Z)$9-dimensional tori driven by irrational rotation on the circle and random composition of $(X,f)$0 area-preserving positive matrices driven by uniquely ergodic subshifts. In these examples, the conditional principle relates leafwise complexity on each fiber to the entropy of invariant measures satisfying the prescribed integral constraint (Liu et al., 2024).

4. Unstable entropy and variational principles conditioned on the unstable foliation

For a $(X,f)$1 partially hyperbolic diffeomorphism $(X,f)$2 with invariant splitting

$(X,f)$3

the unstable foliation $(X,f)$4 permits a metric entropy theory conditioned on unstable leaves. Given a finite partition $(X,f)$5 and an unstable refinement $(X,f)$6, the unstable metric entropy is defined from conditional entropies

$(X,f)$7

The cited work proves that the limit exists and that for any $(X,f)$8 and $(X,f)$9 one has

$(X,d,f)$00

When compared with Ledrappier–Young theory, this quantity coincides with the unstable entropy $(X,d,f)$01 associated with an increasing partition subordinate to $(X,d,f)$02 (Hu et al., 2017).

The leafwise topological counterpart is defined by separated or spanning sets on local unstable plaques. If $(X,d,f)$03 is the maximal cardinality of an $(X,d,f)$04-$(X,d,f)$05-separated set in $(X,d,f)$06, then

$(X,d,f)$07

and

$(X,d,f)$08

The paper also proves

$(X,d,f)$09

where $(X,d,f)$10 is the unstable volume growth along unstable leaves (Hu et al., 2017).

The main variational statement is

$(X,d,f)$11

Its upper bound uses a leafwise Brin–Katok type formula

$(X,d,f)$12

for $(X,d,f)$13-almost every $(X,d,f)$14, while the lower bound constructs empirical measures from maximal $(X,d,f)$15-separated sets inside unstable plaques and applies a Misiurewicz-type averaging argument with conditional entropy estimates (Hu et al., 2017).

This principle is conditional in the precise sense that both entropy notions are restricted to the unstable direction. It isolates complexity caused by the unstable part of a partially hyperbolic system, yields affineness and upper semi-continuity of $(X,d,f)$16, and supplies a Shannon–McMillan–Breiman theorem along unstable leaves. In the Anosov case, where $(X,d,f)$17, it reduces to the classical equality between topological and metric entropy (Hu et al., 2017).

5. Reverse Noether theory and conditional variationality for partial differential equations

In the jet-bundle framework, a second-order system of PDEs is encoded by a source form

$(X,d,f)$18

on $(X,d,f)$19, where $(X,d,f)$20 is a smooth fiber bundle with $(X,d,f)$21 and fiber dimension $(X,d,f)$22. The inverse problem of the calculus of variations asks whether there exists a local Lagrange form $(X,d,f)$23 such that

$(X,d,f)$24

Classically, this is equivalent to the vanishing of the second-order Helmholtz expressions. The cited paper proves a conditional variational principle that deduces this vanishing from symmetry and current hypotheses rather than assuming the Helmholtz conditions a priori (Dafinger, 2019).

The symmetry hypothesis is that the set

$(X,d,f)$25

spans the tangent space pointwise: $(X,d,f)$26 The continuity-equation hypothesis is that for each $(X,d,f)$27 there is a corresponding local continuity equation with characteristic $(X,d,f)$28,

$(X,d,f)$29

Under these two assumptions, Theorem 4.1 concludes that $(X,d,f)$30 is locally variational (Dafinger, 2019).

The proof uses a decomposition of $(X,d,f)$31 involving the Helmholtz operator $(X,d,f)$32 and the “Equation of Continuities and Symmetries”

$(X,d,f)$33

A local linear algebra argument, together with the span condition and the Helmholtz dependencies,

$(X,d,f)$34

forces

$(X,d,f)$35

By the local exactness of the variational sequence, the vanishing of these expressions yields a local Lagrangian (Dafinger, 2019).

The result is local rather than global: “locally variational” means existence of a Lagrange form on $(X,d,f)$36 for each $(X,d,f)$37, while global variationality may be obstructed by cohomology. When the Helmholtz conditions hold, a local Lagrangian can be recovered by the Vainberg–Tonti homotopy,

$(X,d,f)$38

The Lagrangian is unique up to addition of a horizontal divergence term $(X,d,f)$39. The paper also notes that the theorem fails for some third-order source forms, so the conditional principle is specific to second-order source forms in the form proved there (Dafinger, 2019).

6. Random metric spaces, $(X,d,f)$40-valued functionals, and conditional Ekeland principles

Let $(X,d,f)$41 be a probability space and let $(X,d,f)$42 denote the equivalence classes of extended real-valued measurable random variables, ordered by the almost-sure order. A random metric space $(X,d,f)$43 is a set endowed with an $(X,d,f)$44-valued distance, and two natural topologies appear: the $(X,d,f)$45-topology $(X,d,f)$46, which corresponds to convergence in probability, and the stronger locally $(X,d,f)$47 topology $(X,d,f)$48, generated by random radii $(X,d,f)$49. In this setting, one studies proper, lower semicontinuous, lower bounded maps $(X,d,f)$50 (Guo et al., 2011).

The general conditional Ekeland variational principle states that if $(X,d,f)$51 is a $(X,d,f)$52-complete random metric space and $(X,d,f)$53 is proper, $(X,d,f)$54-lower semicontinuous, and lower bounded, then for each $(X,d,f)$55 there exists $(X,d,f)$56 such that

$(X,d,f)$57

and for every $(X,d,f)$58,

$(X,d,f)$59

meaning that there exists $(X,d,f)$60 with $(X,d,f)$61 such that $(X,d,f)$62 on $(X,d,f)$63. The principle is conditional because the penalty, the order, and the notion of strictness are all random and almost-sure (Guo et al., 2011).

A sharper form is proved for complete random normed modules. If $(X,d,f)$64 is $(X,d,f)$65-complete, $(X,d,f)$66 is $(X,d,f)$67-closed, and $(X,d,f)$68 is proper, $(X,d,f)$69-lsc, and lower bounded, then for $(X,d,f)$70 and any $(X,d,f)$71 satisfying

$(X,d,f)$72

there exists $(X,d,f)$73 such that

$(X,d,f)$74

and for every $(X,d,f)$75, $(X,d,f)$76,

$(X,d,f)$77

An analogous $(X,d,f)$78-version requires the countable concatenation property and the local property of the functional. These locality hypotheses are characteristic of conditional analysis on random modules (Guo et al., 2011).

The applications are Bishop–Phelps type density theorems in complete random normed modules under the framework of random conjugate spaces. Support points of a $(X,d,f)$79-closed, $(X,d,f)$80-convex set with the countable concatenation property are $(X,d,f)$81-dense in its $(X,d,f)$82-boundary, and a.s. bounded support functionals are $(X,d,f)$83-dense in the random dual $(X,d,f)$84. A plausible implication is that, in this branch of the literature, a conditional variational principle is not primarily a dynamical entropy statement but a foundational optimization tool for conditional convex analysis and conditional risk measures (Guo et al., 2011).

7. Comparative perspective

Across these works, the term “conditional variational principle” does not refer to a single universal theorem. In the POTP and Pesin settings, it identifies the topological entropy of sets selected by empirical-measure or historic constraints with a supremum of measure entropies over the admissible invariant measures (Yin et al., 2016, Yin et al., 2015). In skew products, the same pattern is refined to fiberwise entropy, conditional level sets, and Legendre-transform formulas relative to a uniquely ergodic base (Liu et al., 2024). In partially hyperbolic dynamics, the conditioning is directional: both metric and topological entropy are taken along unstable manifolds rather than on the full phase space (Hu et al., 2017).

In the PDE setting, the phrase has a different meaning. There the conditional statement is a reverse-Noether theorem: sufficiently many $(X,d,f)$85-projectable symmetries together with matching continuity equations force the Helmholtz conditions and hence local variationality of a second-order source form (Dafinger, 2019). In random analysis, the conditioning is probabilistic and local: the almost-sure order on $(X,d,f)$86, random completeness, and locality conditions replace ordinary scalar order and classical metric structure in Ekeland’s principle (Guo et al., 2011).

This suggests that the common core is structural rather than formal. A variational conclusion is recovered only after restricting the class of admissible objects or the geometry in which comparison is made: invariant measures satisfying a constraint, fibers over almost every base point, unstable leaves, symmetry-current pairs, or random local modules. The phrase therefore functions as a family resemblance term linking entropy theory, inverse problems, and conditional optimization rather than as the name of a single canonical principle.