Entropy Flexibility in Dynamics

• Entropy flexibility is a phenomenon where various entropy invariants (topological, polynomial, persistent) can take any prescribed value across diverse dynamical systems.
• It contrasts with rigidity by demonstrating continuous tunability in some systems while others show fixed, discrete entropy values.
• Techniques such as renewal systems, geometric deformations, and blender-horseshoes are used to construct invariant measures that realize prescribed entropy levels.

Searching arXiv for papers on entropy flexibility and closely related rigidity/flexibility results. Entropy flexibility denotes a family of phenomena in which an entropy quantity can vary continuously, attain all intermediate values, or be prescribed across a broad class of systems, rather than being forced by strong structural constraints. In the literature, this theme appears for topological entropy, measure-theoretic entropy, polynomial entropy, generalized entropy, and Floer-theoretic persistent entropy, and it is repeatedly paired with complementary rigidity statements showing that the same invariant may be constant, discrete, or sharply bounded in more restrictive categories (Arbieto et al., 15 Jul 2025).

1. Formal definitions and entropy functionals

For a homeomorphism $f\colon M\to M$ of a compact metric space, entropy flexibility in discrete time is defined by the requirement that for every

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$

there exist an ergodic invariant measure $\mu_c$ and a compact $f$-invariant set $\Lambda_c\subset M$ such that

$$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$

For a continuous flow $\phi\colon \mathbb R\times M\to M$, the analogous definition requires the same property for every $0\le c<h_{\mathrm{top}}(\phi)$, with $h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$ and $h_\mu(\phi)=h_\mu(\phi_1)$. This notion is explicitly stated to be stronger than the “lowerable” property, because the invariant set must itself be invariant and must carry an ergodic measure with exactly the same entropy (Arbieto et al., 15 Jul 2025).

Polynomial entropy refines zero-topological-entropy dynamics by measuring polynomial orbit-complexity growth. For a continuous map $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$0 on a compact metric space $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$1, if $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$2 denotes the maximal cardinality of an $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$3-separated set, then

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$4

Like topological entropy, $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$5 is a topological conjugacy invariant, but it measures polynomial rather than exponential growth rates (Roth et al., 2021).

Generalized entropy for wandering dynamics is organized in the lattice of growth orders. If $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$6 is the minimal cardinality of an $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$7-generator, then

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$8

and its projections recover classical invariants: $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$9 This places polynomial entropy inside a larger partially ordered space of subexponential and exponential growth types (Correa et al., 2023).

A further entropy-like invariant appears in symplectic topology. For a persistence barcode $\mu_c$0, the Shannon entropy of finite bar lengths is

$\mu_c$1

Persistent entropy is then defined as an asymptotic growth rate of $\mu_c$2 under iteration or action filtration. This suggests that “entropy flexibility” is not tied to a single entropy functional, but to a recurrent realization problem across several entropy theories (Gong, 17 Jun 2026).

2. Rigidity and flexibility in low-dimensional dynamics

The interval is a paradigmatic rigidity setting for polynomial entropy. For continuous $\mu_c$3, the polynomial entropy equals the supremum of the orders of “one-way horseshoes” in $\mu_c$4 or its iterates. A one-way $\mu_c$5-horseshoe is a chain of $\mu_c$6 pairwise disjoint compact sets $\mu_c$7 such that $\mu_c$8 whenever $\mu_c$9, and $f$0 contains a nonrecurrent point. From this characterization one obtains the rigidity statement that if the polynomial entropy of an interval map is finite, then it is an integer; in particular, continuous interval maps satisfy

$f$1

The same work computes the polynomial entropy of all maps in the logistic family and describes the possible values for maps of all Sharkovskii types (Roth et al., 2021).

For piecewise expanding unimodal maps, the situation is flexible rather than discrete. The only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most $f$2, and by the Variational Principle the metric entropy is not larger than the topological entropy. More precisely, for every $f$3 and every $f$4 with $f$5, there exists a piecewise expanding unimodal map $f$6 such that

$f$7

Within the topologically mixing piecewise expanding skew tent family, there is an additional restriction

$f$8

The same setting yields the identity

$f$9

for skew tent maps, presented as a generalization and a different interpretation of the Milnor–Thurston formula (Alsedà et al., 2020).

Boundary maps associated to Fuchsian groups exhibit a sharp rigidity/flexibility split. For generalized Bowen–Series boundary maps $\Lambda_c\subset M$0 on a genus-$\Lambda_c\subset M$1 closed hyperbolic surface, the topological entropy is constant throughout the family: $\Lambda_c\subset M$2 By contrast, for extremal or short-cycle parameters admitting a unique smooth invariant probability $\Lambda_c\subset M$3, the measure-theoretic entropy is

$\Lambda_c\subset M$4

and it attains every value in $\Lambda_c\subset M$5, with the maximum at the regular fundamental $\Lambda_c\subset M$6-gon (Abrams et al., 2021).

Negative-curvature surfaces supply an analogous dichotomy for geodesic flows. With fixed total area $\Lambda_c\subset M$7 and $\Lambda_c\subset M$8, the set of all achievable pairs $\Lambda_c\subset M$9 for smooth negatively curved metrics is exactly the open region

$$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$0

together with the critical corner $$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$1. Thus the only obstructions are the Katok inequalities. Inside a fixed conformal class with nonpositive curvature and fixed area, there is a different pattern: the diameter is bounded above, the Laplace spectrum is bounded below away from zero, yet for every $$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$2 there exists a negatively curved metric with

$$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$3

A plausible implication is that entropy flexibility can coexist with substantial geometric rigidity when the conformal background is fixed (Erchenko et al., 2017, Barthelmé et al., 2017).

3. Full realization on continua, dendrites, and wandering systems

For polynomial entropy on continua, full flexibility is available. For every $$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$4, there exists a continuum $$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$5 and a homeomorphism $$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$6 with $$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$7. The construction proceeds by first realizing a dense set of positive values on compact metric spaces, then taking a disjoint union plus one extra fixed point so that the entropy is the supremum of the chosen values, and finally passing to a cone over the resulting space; polynomial entropy is unchanged by products with identity and by factors (Roth et al., 2021).

The same paper gives partial flexibility results on dendrites. By embedding symbolic subshifts into dynamics on the Gehman dendrite $$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$8, one obtains maps $$h_{\mathrm{top}}(f|_{\Lambda_c}) \;=\; h_{\mu_c}(f) \;=\; c.$$9 satisfying

$\phi\colon \mathbb R\times M\to M$0

From this one deduces that for every $\phi\colon \mathbb R\times M\to M$1 there is a surjective dendrite map with $\phi\colon \mathbb R\times M\to M$2, and by more elaborate branch-order arguments one can reach all $\phi\colon \mathbb R\times M\to M$3. It remains open whether values in $\phi\colon \mathbb R\times M\to M$4 can be realized on dendrites (Roth et al., 2021).

A stronger contrast appears for pointwise periodic systems. For every real number $\phi\colon \mathbb R\times M\to M$5, there exists a compact connected metric space $\phi\colon \mathbb R\times M\to M$6 and a pointwise periodic homeomorphism $\phi\colon \mathbb R\times M\to M$7 such that

$\phi\colon \mathbb R\times M\to M$8

In particular, there are pointwise periodic homeomorphisms with positive polynomial entropy. This contrasts with Montgomery’s theorem for connected manifolds, where pointwise periodic implies globally periodic, and with the zero-polynomial-entropy result for local dendrites (Đorić et al., 26 Jun 2026).

Generalized entropy for wandering dynamics yields a different realization theorem. Let $\phi\colon \mathbb R\times M\to M$9 be the class of sphere homeomorphisms whose non-wandering set consists of only one fixed point. The flexibility theorem states: if $0\le c<h_{\mathrm{top}}(\phi)$0 satisfies either $0\le c<h_{\mathrm{top}}(\phi)$1 or $0\le c<h_{\mathrm{top}}(\phi)$2, then there exists $0\le c<h_{\mathrm{top}}(\phi)$3 such that $0\le c<h_{\mathrm{top}}(\phi)$4. The corollary gives explicit examples $0\le c<h_{\mathrm{top}}(\phi)$5 with

$0\le c<h_{\mathrm{top}}(\phi)$6

hence $0\le c<h_{\mathrm{top}}(\phi)$7 while $0\le c<h_{\mathrm{top}}(\phi)$8. This shows that generalized entropy refines polynomial entropy even inside a fixed polynomial-entropy class (Correa et al., 2023).

4. Hyperbolic, partially hyperbolic, and generic smooth settings

Suspension flows over shifts of finite type provide an explicit entropy-flexible class in the formal sense. If $0\le c<h_{\mathrm{top}}(\phi)$9 is a subshift of finite type and $h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$0 is a continuous roof function, then its suspension flow $h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$1 satisfies: for every

$h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$2

there is a compact $h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$3-invariant subset $h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$4 such that $h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$5 is strictly ergodic and

$h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$6

where $h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$7 is its unique invariant measure. The proof uses renewal systems techniques and Abramov’s formula (Arbieto et al., 15 Jul 2025).

The same framework establishes genericity. Entropy flexibility is a $h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$8-generic property for flows generated by $h_{\mathrm{top}}(\phi)=h_{\mathrm{top}}(\phi_1)$9 vector fields on any closed three-dimensional manifold and for homeomorphisms of any compact surface. It is also proved for star flows and for asymptotically sectional-hyperbolic attracting sets, yielding entropy flexibility for every Anosov or Axiom-A flow, every Lorenz-like or sectional-hyperbolic attractor, every multi-singular-hyperbolic or star flow, and related examples (Arbieto et al., 15 Jul 2025).

Partially hyperbolic systems support a finer joint flexibility theorem involving entropy and a Lyapunov exponent. Suppose $h_\mu(\phi)=h_\mu(\phi_1)$0 is a $h_\mu(\phi)=h_\mu(\phi_1)$1 partially hyperbolic diffeomorphism with one-dimensional center,

$h_\mu(\phi)=h_\mu(\phi_1)$2

and assume minimality of the strong stable and unstable foliations on a transitive compact set $h_\mu(\phi)=h_\mu(\phi_1)$3, together with a pair of blender-horseshoes, one center contracting and one center expanding. Writing

$h_\mu(\phi)=h_\mu(\phi_1)$4

the theorem states that for every $h_\mu(\phi)=h_\mu(\phi_1)$5 and every $h_\mu(\phi)=h_\mu(\phi_1)$6, there exists an ergodic $h_\mu(\phi)=h_\mu(\phi_1)$7 with

$h_\mu(\phi)=h_\mu(\phi_1)$8

This is full entropy–exponent flexibility among ergodic measures with fixed center exponent (Díaz et al., 6 May 2025).

A related program concerns entropy and Lyapunov exponents for conservative Anosov systems. Examples are constructed of derived from Anosov diffeomorphisms with the metric entropy larger than the entropy of the linear part, and of Anosov diffeomorphisms with the strong unstable exponent larger than the strong unstable exponent of the linear part. These constructions rely on a new type of deformation that goes beyond the previous Shub-Wilkinson and Baraviera-Bonatti techniques (Carrasco et al., 2021).

5. Entropy under orbit equivalence and in Floer-theoretic persistence

Quantitative orbit equivalence yields a threshold phenomenon for entropy preservation. It follows from the work of Kerr and Li that if the cocycles of an orbit equivalence are $h_\mu(\phi)=h_\mu(\phi_1)$9-integrable, the entropy is preserved. The odomutant construction shows that this result is optimal: there are odomutants of all positive entropies orbit equivalent to an odometer, with almost $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$00-integrable cocycles. More precisely, for every $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$01, there exists a Cantor-minimal homeomorphism $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$02 with

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$03

that is strongly orbit-equivalent to an odometer via cocycles in $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$04. The same paper proves that even Kakutani equivalence and $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$05 orbit equivalence are not the same by producing a non-loosely Bernoulli system of Feldman that is an odomutant (Correia, 2 Apr 2025).

Persistent entropy of Floer persistence barcodes introduces an entropy invariant based on the Shannon entropy of finite bar lengths. For Hamiltonian diffeomorphisms, the relative and absolute persistent entropies coincide with the corresponding barcode entropies. For Liouville domains there are general comparison inequalities,

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$06

and equality holds when $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$07. A subexponential length-growth criterion gives equality beyond the vanishing-symplectic-homology case. For cotangent disk bundles of negatively curved manifolds,

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$08

The paper also formulates flexibility and rigidity-type questions for barcode and persistent entropies of Reeb flows (Gong, 17 Jun 2026).

These results suggest that entropy flexibility can be studied not only by varying a dynamical system inside a category, but also by varying the equivalence relation, filtration, or coding through which entropy is measured. In some settings the entropy survives such transformations exactly; in others, the sharp boundary is the integrability class of the cocycle or the growth rate of barcode lengths.

6. Entropy as a mechanism of physical flexibility

Outside smooth and topological dynamics, entropy often appears as the mechanism that stabilizes flexible configurations. In mobile DNA-coated colloids, free energy calculations show that at coexistence the floppy square and CsCl crystals satisfy

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$09

so that $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$10 relative to the compact crystals. The stabilizing contribution is vibrational entropy, amplified by low-frequency “floppy modes” of the dynamical matrix. Remarkably, these floppy phases survive even in the formal limit $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$11, showing purely entropic stabilization (Hu et al., 2017).

Zeolite frameworks display a related configurational effect. Treating idealized frameworks as periodic mechanical trusses, the number of flexible folding mechanisms is strongly peaked at the minimum density end of the flexibility window. Twenty-five of the 197 known zeolite frameworks exhibit an extensive flexibility, where the number of unique mechanisms increases linearly with the volume when long wavelength mechanisms are included. Extensively flexible frameworks therefore have a maximum in configurational entropy, as large crystals, at their lowest density (Kapko et al., 2011).

For the flexible q-TIP4P/F water model, the potential-energy-landscape formalism shows that the configurational entropy $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$12 is surprisingly similar to that reported previously for rigid water models, suggesting that intramolecular flexibility does not necessarily add roughness to the PEL. The Adam–Gibbs relation

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$13

holds for the flexible model. At the same time, classical high-frequency modes produce an unphysical negative vibrational entropy, whereas the PEL-based configurational entropy remains positive and consistent with Gaussian-landscape expectations (Eltareb et al., 2024).

A nonequilibrium cell-physics formulation reaches a closely related conclusion. Modeling sarcomere-length variability with a Fokker–Planck equation, the Shannon entropy of the steady-state length distribution is

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$14

Enforcing nonnegative total entropy production yields

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$15

so increased entropy lowers the maximum allowable binding energy and lowers the energy barrier for remodeling. The ordered sarcomere arrangements in muscle-type cells correspond to higher binding energies and more stable cytoskeletal configurations, whereas the increased entropy associated with the inherent randomness of sarcomere structures in nonmuscle-type cells enables flexible adaptation to environmental demands (Ueda et al., 2024).

In control of distributed loads, entropy is used to quantify aggregate flexibility directly. The maximum-entropy feedback is defined by maximizing total conditional entropy subject to feasibility, and its unique analytic form is

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$16

This feedback can be approximated by reinforcement learning and inserted as a penalty term in penalized predictive control,

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$17

On an adaptive electric-vehicle charging dataset, PPC is reported to outperform classical MPC (Li et al., 2020).

7. Open problems, thresholds, and recurring structures

Several papers identify explicit frontiers where flexibility is not yet understood. For dendrites, values in $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$18 are still open for polynomial entropy realization (Roth et al., 2021). For generalized entropy of wandering sphere homeomorphisms, it remains open whether one can drop the LIP assumption altogether and realize every $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$19 (Correa et al., 2023). For Liouville domains, it is asked whether there exists a strict gap

$0\;\le\;c\;<\;h_{\mathrm{top}}(f)$20

when $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$21, and whether rigidity at the Katok threshold persists for all area-one metrics on surfaces (Gong, 17 Jun 2026).

A recurring structural pattern is the coexistence of realization theorems with exact threshold theorems. On the interval, polynomial entropy is rigidly integer-valued, whereas on continua every value in $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$22 is realized (Roth et al., 2021). For generalized Bowen–Series maps, topological entropy is constant while smooth entropy fills an interval (Abrams et al., 2021). For quantitative orbit equivalence, $0\;\le\;c\;<\;h_{\mathrm{top}}(f)$23-integrability is the sharp preservation threshold for entropy, while below that threshold one can realize all positive entropies by odomutants orbit equivalent to an odometer (Correia, 2 Apr 2025). For suspension flows, hyperbolic-like flows, and generic low-dimensional smooth systems, every intermediate entropy can be attained on a strictly ergodic subsystem (Arbieto et al., 15 Jul 2025).

This body of work suggests that entropy flexibility is best understood not as a single theorem, but as a comparative principle. In some categories entropy behaves as a rigid moduli-type invariant; in others it is a tunable quantity controlled by horseshoes, renewal systems, special flows, blender-horseshoes, orbit-equivalence cocycles, or geometric deformations. The technical content of the subject lies in determining which structural hypotheses force rigidity and which permit complete realization.