Bowen Topological Entropy

• Bowen topological entropy is a dimension-like invariant measuring the complexity of orbit segments on compact metric spaces via spanning and separated sets.
• It employs Carathéodory coverings with variable-length Bowen balls to quantify the complexity of arbitrary, including noncompact, subsets.
• Extensions of Bowen entropy include nonautonomous systems, amenable group actions, and multifractal analyses, linking topological and measure-theoretic approaches.

Bowen topological entropy is a family of closely related invariants originating in Bowen’s 1973 work on dynamical complexity. In the narrowest usage, it is the metric entropy of a continuous map on a compact metric space defined through Bowen–Dinaburg metrics, spanning sets, or separated sets. In a broader and now standard usage, it is the Carathéodory-type entropy of an arbitrary subset $Z$ obtained by covering $Z$ with Bowen balls of variable lengths and extracting the critical exponential covering rate. The literature also extends Bowen entropy to nonautonomous systems, flows, amenable group actions, partially hyperbolic unstable foliations, and multifractal variants, while preserving the central idea that orbit segments, rather than single iterates, determine complexity (Caldas et al., 2015, Zheng et al., 2014, Ju, 31 Dec 2025).

1. Classical formulations and terminological distinctions

For a continuous map $f:X\to X$ on a metric space $(X,d)$, the Bowen metric is

$d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$

and the $(n,\varepsilon)$-Bowen ball is

$$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$

A set is $(n,\varepsilon)$-spanning if every point is $d_n$-close to some point of the set, and $(n,\varepsilon)$-separated if any two distinct points are $Z$0-apart by at least $Z$1. Bowen’s metric entropy is then

$Z$2

where $Z$3 is the minimal cardinality of an $Z$4-spanning set and $Z$5 is the maximal cardinality of an $Z$6-separated set (1711.02562, Caldas et al., 2015).

On compact metric spaces this metric entropy agrees with the usual topological entropy. In particular, the Bowen–Dinaburg–Goodman variational principle yields

$Z$7

for any compatible metric $Z$8 (1711.02562). This agreement underlies the common habit of referring to $Z$9 itself as Bowen topological entropy in compact settings, including work on generalized Bowen–Series boundary maps, where the entropy is computed through separated sets on the compact circle rather than through subset entropy (Abrams et al., 2021).

A persistent terminological ambiguity is that “Bowen topological entropy” may denote either the global metric entropy of a map on the whole compact space or Bowen’s entropy of a subset $f:X\to X$0. The distinction matters because the subset theory is genuinely Carathéodory-dimensional and is designed for noncompact, noninvariant, or dynamically exceptional sets (Zheng et al., 2014, Pacifico et al., 2019).

2. Bowen entropy of subsets and the Carathéodory construction

For a continuous map $f:X\to X$1 on a compact metric space and a subset $f:X\to X$2, Bowen’s subset entropy is defined by covering $f:X\to X$3 with Bowen balls of variable lengths. One standard form is

$f:X\to X$4

followed by

$f:X\to X$5

and then the critical exponent

$f:X\to X$6

This is the standard Bowen entropy of possibly noncompact sets for maps (Ju, 31 Dec 2025, Pacifico et al., 2019, Tian et al., 2018, Lin et al., 25 Aug 2025).

The construction is formally analogous to Hausdorff dimension. The weights $f:X\to X$7 play the role of scale weights, but the scales are dynamical lengths measured by orbit agreement rather than geometric diameters. This is why Bowen entropy is often described as a dimension-like invariant of subsets (Zheng et al., 2014, Dou et al., 2017). The analogy becomes especially explicit in amenable group actions, where Bowen entropy is defined by weights $f:X\to X$8 over Bowen sets indexed by Følner sets, and in nonautonomous systems, where variable-length strings or Bowen balls play the same role (Zheng et al., 2014, Sarkooh, 2022).

Several structural properties survive across these formulations. Bowen entropy is monotone in the subset, and in many settings it is countably stable: $f:X\to X$9 (Tian et al., 2018, Pacifico et al., 2019). For compact $(X,d)$0-invariant sets it agrees with fixed-length capacity-style formulations, whereas for arbitrary subsets the variable-length covering formulation is essential (Tian et al., 2018, Ju, 31 Dec 2025). This distinction is precisely what allows Bowen entropy to detect complexity of irregular, generic, saturated, or leafwise-defined sets that are invisible to invariant-set entropy alone.

3. Variational principles, local entropy, and generic points

A major line of development identifies Bowen entropy with measure-theoretic local entropy through variational principles. For amenable group actions, Bowen entropy on a compact subset $(X,d)$1 satisfies

$(X,d)$2

under the growth condition $(X,d)$3 (Zheng et al., 2014). In the same setting, for an ergodic invariant measure $(X,d)$4 and a tempered Følner sequence satisfying that growth condition, the set $(X,d)$5 of $(X,d)$6-generic points has Bowen entropy exactly equal to the measure entropy: $(X,d)$7 (Zheng et al., 2016). This extends Bowen’s classical $(X,d)$8-action result to countable amenable groups.

For continuous flows, a continuous-time Carathéodory construction yields an entropy $(X,d)$9 on arbitrary subsets $d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$0, and the paper on flows proves the exact time-one reduction

$d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$1

more generally $d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$2 for $d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$3 (Pacifico et al., 2019). In that setting Bowen’s inequality takes the form

$d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$4

with equality for ergodic $d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$5 (Pacifico et al., 2019). Thus the entropy of generic points remains governed by local ball-measure asymptotics, exactly as in discrete time.

Nonautonomous systems admit parallel results. For an NDS $d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$6, the Bowen metric is

$d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$7

with Bowen balls $d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$8, and the subset entropy is again defined by a Carathéodory covering construction (Sarkooh, 2022, Ju, 31 Dec 2025). For nonempty compact $d_n(x,y)=\max_{0\le k\le n-1} d(f^k(x),f^k(y)),$9,

$(n,\varepsilon)$0

where $(n,\varepsilon)$1 is defined from lower local entropy via the nonautonomous Bowen balls (Sarkooh, 2022). A neutralized variant replaces $(n,\varepsilon)$2 by $(n,\varepsilon)$3; the resulting neutralized Bowen entropy and neutralized weighted Bowen entropy coincide and satisfy a corresponding variational principle in terms of lower neutralized Brin–Katok local entropy and neutralized Katok entropy (Sarkooh et al., 2023).

These results collectively show that Bowen entropy is not merely a topological counting invariant. It is the subset-level counterpart of local entropy formulas, and the bridge between the topological size of exceptional sets and the pointwise decay of invariant or noninvariant measures on Bowen balls.

4. Extensions: nonautonomous, intermediate, unstable, and multifractal versions

A recent nonautonomous refinement introduces a one-parameter family of intermediate topological entropies that interpolate between Bowen entropy and capacity entropies. For $(n,\varepsilon)$4, admissible covers are restricted so that all strings in a single cover have lengths satisfying

$(n,\varepsilon)$5

At $(n,\varepsilon)$6 there is no restriction and one recovers Bowen entropy; at $(n,\varepsilon)$7 only same-length covers are allowed, recovering lower and upper capacity topological entropies (Ju, 31 Dec 2025). The resulting lower and upper intermediate entropies satisfy, for $(n,\varepsilon)$8,

$(n,\varepsilon)$9

and similarly for the lower version, implying continuity on $$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$0 and possible discontinuity at $$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$1 (Ju, 31 Dec 2025). In the example built from the shift on $$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$2, the intermediate entropies equal $$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$3 for every $$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$4 while the Bowen entropy at $$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$5 is $$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$6, making the discontinuity explicit (Ju, 31 Dec 2025).

For partially hyperbolic systems, Tian and Wu define unstable Bowen topological entropy by localizing Bowen’s Carathéodory construction to unstable leaves. If $$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$7 denotes an unstable Bowen ball in the local unstable manifold, then the unstable entropy of $$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$8 is obtained from covers by such sets and can equivalently be written as

$$B_n(x,\varepsilon)=\{y\in X:d_n(x,y)<\varepsilon\}.$$9

(Tian et al., 2018). They prove an unstable entropy distribution principle and a variational principle

$(n,\varepsilon)$0

for nonempty compact $(n,\varepsilon)$1 (Tian et al., 2018). This places Bowen entropy directly inside dimension theory and multifractal analysis for partially hyperbolic dynamics.

A different multifractal extension introduces $(n,\varepsilon)$2-Bowen and $(n,\varepsilon)$3-packing topological entropies. At $(n,\varepsilon)$4, the paper states that the classical Bowen and packing entropies are recovered: $(n,\varepsilon)$5 in the notation of that work (Wang et al., 28 Feb 2025). The same paper develops measurability properties of these multifractal entropies on hyperspaces and relates $(n,\varepsilon)$6-packing entropy to the topological entropy of local-entropy level sets through a Legendre-transform framework (Wang et al., 28 Feb 2025).

5. Amenable actions, noncompact spaces, and generalized sequence-space formulations

For countable amenable group actions on compact metric spaces, Bowen entropy is defined along a fixed Følner sequence through Bowen metrics

$(n,\varepsilon)$7

and Carathéodory sums weighted by $(n,\varepsilon)$8 (Zheng et al., 2014, Dou et al., 2017, Zheng et al., 2016). Under temperedness and the condition

$(n,\varepsilon)$9

the Bowen entropy of the whole space equals the classical topological entropy of the action (Zheng et al., 2014). A later paper gives a purely topological proof for tempered Følner sequences, dispensing with the extra growth condition in the whole-space equality and also proving that, for amenable subshifts equipped with the natural Følner-based metric, Hausdorff dimension equals topological entropy (Dou et al., 2017).

Noncompact settings require a different caution. In locally compact separable metrizable systems, Bowen metric entropy generally depends on the compatible metric, and the canonical topological entropy is recovered as the minimum over compatible metrics: $d_n$0 for any continuous map $d_n$1 (Caldas et al., 2015). This resolves the compact/noncompact asymmetry: on compact spaces $d_n$2 is canonical, whereas on noncompact spaces a fixed choice of $d_n$3 may overestimate the intrinsic topological entropy (Caldas et al., 2015). The Lie-group computation paper sharpens this point by observing that on noncompact Lie groups Bowen’s metric entropy $d_n$4 attached to a left-invariant distance can be strictly larger than the actual topological entropy $d_n$5; for $d_n$6 on $d_n$7, the paper states $d_n$8 while $d_n$9 (1711.02562).

A further abstraction replaces orbit sets of a single map by arbitrary subsets of infinite product spaces. For $(n,\varepsilon)$0, generalized topological entropy is defined by covering $(n,\varepsilon)$1 with $(n,\varepsilon)$2-cylinders coming from open covers of $(n,\varepsilon)$3, and on compact metric spaces this generalized entropy coincides with Bowen $(n,\varepsilon)$4-entropy (Sadr et al., 2020). For a continuous map $(n,\varepsilon)$5, if $(n,\varepsilon)$6, then

$(n,\varepsilon)$7

so classical topological entropy is recovered as the sequence-space entropy of the orbit graph (Sadr et al., 2020). This formulation isolates the orbit-complexity mechanism behind Bowen’s construction and makes it available for arbitrary sequence sets and even self-similar coding structures.

6. Computations, rigidity phenomena, and conceptual scope

Bowen entropy has proved effective both in explicit computations and in rigidity results. For generalized Bowen–Series boundary maps $(n,\varepsilon)$8 associated to cocompact torsion-free Fuchsian groups of genus $(n,\varepsilon)$9, the classical topological entropy on the compact boundary circle is rigid across all parameters $Z$00 and equals

$Z$01

(Abrams et al., 2021). Here “Bowen topological entropy” refers to the standard compact-space entropy of the map $Z$02, not subset entropy (Abrams et al., 2021).

In Lie-group dynamics, the exact topological entropy of a continuous endomorphism $Z$03 of a Lie group $Z$04 is carried entirely by the maximal torus in the center of the maximal connected $Z$05-invariant subgroup: $Z$06 (1711.02562). This shows that Bowen’s derivative-based metric formula on a fixed left-invariant metric is not the final answer in the noncompact case; entropy localizes on a compact central torus (1711.02562).

In complex dynamics, Bowen’s definition on noncompact spaces is used to prove that every entire transcendental function has infinite topological entropy: $Z$07 (Wendt, 2020). The same paper proves that for a meromorphic map without wandering domains, the entropy of the Fatou set vanishes and all entropy is concentrated on the Julia set (Wendt, 2020). These results show that Bowen entropy remains meaningful beyond compact or uniformly hyperbolic settings, provided one uses the open-cover or subset-based formulation appropriate to noncompact dynamics.

A common misconception is to treat all notions called “Bowen entropy” as interchangeable. The literature instead supports a three-way distinction. First, on compact metric spaces, Bowen’s separated/spanning-set entropy is the classical topological entropy. Second, Bowen’s subset entropy is a Carathéodory invariant for arbitrary sets and is indispensable for irregular, generic, or multifractal sets. Third, on noncompact spaces, fixed-metric Bowen entropy may be only an upper bound for the canonical topological entropy unless one minimizes over compatible metrics or restricts to the appropriate compact recurrent part (Caldas et al., 2015, 1711.02562). Recognizing which of these roles is in play is essential for reading contemporary work on Bowen topological entropy.

Taken together, the modern theory presents Bowen topological entropy as a unifying orbit-complexity formalism. It governs whole-space entropy on compact systems, measures the size of exceptional subsets through variable-length coverings, extends to group actions and nonautonomous dynamics, admits local-entropy variational principles, and supports refined constructions such as intermediate, unstable, neutralized, and multifractal entropies (Ju, 31 Dec 2025, Sarkooh, 2022, Sarkooh et al., 2023, Wang et al., 28 Feb 2025).