Nonautonomous Symbolic Systems

• Nonautonomous symbolic systems are defined by time-dependent spaces, maps, and alphabets, providing a framework to model evolving dynamics.
• Their entropy and pressure theories use liminf/limsup approaches and admissible cover constraints to avoid spurious complexity from refining partitions.
• Explicit constructions such as Bernoulli measures and equiconjugacy ensure these systems are robust and serve as key examples in nonautonomous thermodynamic formalism.

Nonautonomous symbolic systems are symbolic dynamical systems in which the symbolic space, the shift-like map, or both vary with time. In the standard process formulation, a nonautonomous dynamical system is a sequence $$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty}$$ with continuous maps $T_k:X_k\to X_{k+1}$; symbolic realizations take the $X_k$ to be spaces of admissible tails, often with changing alphabets or time-dependent transition rules, and the $T_k$ to be left shifts or coding maps. Unlike autonomous subshifts, this setting replaces a single phase space, a single transformation, and a single invariant measure by sequences, and it requires entropy and pressure theories that use $\limsup$ or $\liminf$ together with admissibility restrictions on covers or partitions in order to exclude spurious complexity created purely by time-dependent observation scales (Kawan, 2013, Kawan, 2017, Chen et al., 2 Aug 2025, Chen et al., 14 Sep 2025).

1. Formal models and symbolic realizations

The modern theory developed in stages. Kawan introduced metric entropy for nonautonomous dynamical systems as a generalization of the Kolmogorov–Sinai notion and related it to nonautonomous topological entropy by a one-sided variational inequality (Kawan, 2013). Later work introduced a family of topological pressures on subsets of general nonautonomous dynamical systems, and then treated nonautonomous symbolic systems and strongly uniformly expansive systems as central examples and coding targets (Chen et al., 2 Aug 2025, Chen et al., 14 Sep 2025).

A nonautonomous dynamical system may be written as

$$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$

with iterates

$$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$

Equivalent notation appears in the entropy literature as $(X_{1,\infty},f_{1,\infty})$, with

$f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$

The essential point is the same in both formulations: time is indexed explicitly, the state space may change with time, and the evolution law may also change with time.

The explicit full symbolic model fixes a sequence of alphabet sizes $T_k:X_k\to X_{k+1}$0 with $T_k:X_k\to X_{k+1}$1 and defines

$T_k:X_k\to X_{k+1}$2

together with the left shifts

$T_k:X_k\to X_{k+1}$3

The resulting system $T_k:X_k\to X_{k+1}$4 is the nonautonomous symbolic system, and when $T_k:X_k\to X_{k+1}$5 it reduces to the usual one-sided full shift on $T_k:X_k\to X_{k+1}$6 symbols (Chen et al., 14 Sep 2025).

This full-shift model is only the most explicit case. A natural symbolic specialization also covers spaces $T_k:X_k\to X_{k+1}$7 determined by changing alphabets $T_k:X_k\to X_{k+1}$8, time-dependent transition rules $T_k:X_k\to X_{k+1}$9, or nonstationary Markov constraints, with shift-like maps $X_k$0 (Kawan, 2017). In this sense, nonautonomous symbolic systems include nonstationary subshifts, time-varying shifts of finite type, and symbolic codings of time-dependent expanding systems.

2. Geometric structure and entropy foundations

The symbolic spaces $X_k$1 are compact ultrametric spaces. Their cylinders

$X_k$2

are clopen and form a basis, and the ultrametric has the net property: two cylinders are either disjoint or one contains the other. With

$X_k$3

the $X_k$4-step Bowen balls have exact cylinder descriptions:

$X_k$5

and

$X_k$6

Thus cylinders are precisely open and closed Bowen balls (Chen et al., 14 Sep 2025).

Entropy theory in the nonautonomous setting differs decisively from the autonomous case. On the topological side, one may define entropy from refined open covers or from spanning and separated sets, but one does not take the supremum over all cover sequences. Kawan’s framework restricts to sequences of open covers whose Lebesgue numbers are bounded away from zero, because otherwise entropy can be artificially forced to be $X_k$7 by choosing covers whose diameters shrink exponentially with time (Kawan, 2017). On the measure-theoretic side, the basic partition entropy is

$X_k$8

but one cannot simply take the supremum over all partition sequences, because even the identity system can be made to have arbitrarily large entropy by choosing partitions that refine too rapidly in time (Kawan, 2013).

For that reason, Kawan introduced admissible classes of partition sequences. An admissible class $X_k$9 is nonempty and satisfies a uniform cardinality bound, closure under coarsening, and closure under dynamical block refinements

$T_k$0

Metric entropy relative to $T_k$1 is then defined by

$T_k$2

This formalism provides the measure-theoretic counterpart needed for symbolic systems with time-dependent observations, and it satisfies invariance under appropriate isomorphisms, a power rule, and a Rokhlin-type continuity estimate (Kawan, 2013).

A natural symbolic specialization uses one-cylinder partitions. In that case, the joined partition

$T_k$3

is the partition by admissible words of length $T_k$4, so the corresponding Shannon entropy is the entropy of the induced distribution on length-$T_k$5 cylinders. This gives the expected interpretation of metric entropy as asymptotic information per symbol (Kawan, 2013).

The bridge to topological entropy is the Misiurewicz class $T_k$6. A partition sequence belongs to $T_k$7 if each atom admits compact cores with arbitrarily small measure loss and a uniform separation constant. Under equicontinuity, $T_k$8 is admissible and satisfies the one-sided variational inequality

$T_k$9

The full nonautonomous variational principle remains open in general, but zero-dimensional spaces are a particularly favorable case for the fine-scale partition theory, and symbolic spaces are often zero-dimensional (Kawan, 2013, Kawan, 2017).

3. Pressure theory on symbolic spaces

The 2025 symbolic theory is organized around four pressure notions on subsets $\limsup$0: lower and upper capacity pressures $\limsup$1, Bowen pressure $\limsup$2, and packing pressure $\limsup$3. The more general topological framework also includes spanning versions $\limsup$4, and for equicontinuous potentials the spanning and separated formulations coincide (Chen et al., 2 Aug 2025).

For a potential sequence

$\limsup$5

the nonautonomous Birkhoff sums are

$\limsup$6

In symbolic systems, the exact cylinder geometry makes the pressure formulas explicit. If $\limsup$7 and

$\limsup$8

then for equicontinuous $\limsup$9 the lower and upper capacity pressures are given by liminf and limsup growth rates of weighted sums over admissible words:

$\liminf$0

$\liminf$1

where $\liminf$2, and the limits are independent of the representatives (Chen et al., 14 Sep 2025).

A notable homogeneous property is that nonempty interior already carries full pressure. For

$\liminf$3

if $\liminf$4 has nonempty interior, then

$\liminf$5

for every equicontinuous $\liminf$6. For nonempty open compact $\liminf$7, one also has

$\liminf$8

The proof uses the cylinder structure and equiconjugacies between subsystems carried by cylinders of equal rank (Chen et al., 14 Sep 2025).

The central regularity condition for explicit formulas is strongly bounded variation. Writing

$\liminf$9

the condition requires a constant $$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$0 such that for every $$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$1, every $$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$2, and all $$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$3,

$$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$4

Under this hypothesis,

$$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$5

for

$$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$6

Thus general symbolic potentials reduce to first-coordinate potentials (Chen et al., 14 Sep 2025).

If the potential depends only on the first coordinate,

$$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$7

then the partition function factors:

$$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$8

The pressure formulas become

$$(\boldsymbol X,\boldsymbol T)=\{(X_k,T_k)\}_{k=0}^{\infty},$$9

and

$$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$0

Under strongly bounded variation, the same formulas remain valid for general continuous symbolic potentials, with arbitrary

$$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$1

For entropy, that is, for $$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$2, these reduce to

$$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$3

$$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$4

These are the exact nonautonomous analogues of the autonomous full-shift formulas (Chen et al., 14 Sep 2025).

4. Measure sequences, Bernoulli measures, and equilibrium states

The measure-theoretic side of nonautonomous symbolic dynamics replaces a single invariant measure by an invariant measure sequence. In Kawan’s notation, this means a sequence $$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$5 satisfying

$$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$6

or equivalently, in the symbolic shift setting,

$$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$7

This is the natural nonautonomous analogue of a shift-invariant measure (Kawan, 2017).

For the explicit full symbolic systems of the 2025 theory, the principal model is the nonautonomous Bernoulli measure. Given positive probability vectors

$$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$8

the level-$$T_k^j=T_{k+j-1}\circ\cdots\circ T_k,\qquad T_k^0=\mathrm{id}_{X_k}.$$9 measure is defined on cylinders by

$(X_{1,\infty},f_{1,\infty})$0

This is the natural time-dependent product measure (Chen et al., 14 Sep 2025).

For bounded alphabets, the lower and upper entropies of such a Bernoulli measure are

$(X_{1,\infty},f_{1,\infty})$1

$(X_{1,\infty},f_{1,\infty})$2

For first-coordinate potentials $(X_{1,\infty},f_{1,\infty})$3, the integrated lower and upper pressures take the expected entropy-plus-energy form:

$(X_{1,\infty},f_{1,\infty})$4

$(X_{1,\infty},f_{1,\infty})$5

The uniform Bernoulli choice $(X_{1,\infty},f_{1,\infty})$6 yields a measure of maximal lower and upper entropy (Chen et al., 14 Sep 2025).

The symbolic subset theory also has explicit variational principles. For compact $(X_{1,\infty},f_{1,\infty})$7,

$(X_{1,\infty},f_{1,\infty})$8

for equicontinuous $(X_{1,\infty},f_{1,\infty})$9, and under extra boundedness assumptions,

$f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$0

This sharpens the more general situation in which a full nonautonomous variational principle remains open (Kawan, 2017, Chen et al., 14 Sep 2025).

For strongly bounded variation potentials, equilibrium states are explicit. Choose

$f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$1

and define

$f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$2

Then the associated nonautonomous Bernoulli measure is both a Bowen equilibrium state and a packing equilibrium state on the full nonautonomous shift. Moreover, if $f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$3 is compact and has positive mass for this reference Bernoulli measure, then the normalized restriction to $f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$4 is again an equilibrium state on $f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$5 (Chen et al., 14 Sep 2025).

5. Structural properties, recodings, and symbolic extensions

The subset-pressure framework satisfies a collection of structural properties that make it robust under symbolic constructions. For

$f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$6

the pressures are monotone in the subset $f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$7, and the comparison inequalities

$f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$8

hold on arbitrary subsets. Bowen and packing pressures are countably stable, while the lower and upper capacity-type quantities behave more like box capacities. The map $f_k^n=f_{k+n-1}\circ\cdots\circ f_k.$9 is Lipschitz in the sup norm, and if two potentials agree from some time onward, then their pressures agree. These properties are part of the general nonautonomous thermodynamic formalism rather than symbolic phenomena alone, but they are directly usable in symbolic applications (Chen et al., 2 Aug 2025).

Uniform symbolic recoding is handled by equiconjugacy invariance. If $T_k:X_k\to X_{k+1}$00 is an equiconjugacy between nonautonomous systems, then for equicontinuous $T_k:X_k\to X_{k+1}$01,

$T_k:X_k\to X_{k+1}$02

for all six pressures. This gives the expected invariance under uniformly continuous time-dependent block codes and other symbolic recodings (Chen et al., 2 Aug 2025).

Higher-block presentations are governed by power rules. If

$T_k:X_k\to X_{k+1}$03

and

$T_k:X_k\to X_{k+1}$04

then under equicontinuity of $T_k:X_k\to X_{k+1}$05,

$T_k:X_k\to X_{k+1}$06

for

$T_k:X_k\to X_{k+1}$07

and also for $T_k:X_k\to X_{k+1}$08 if $T_k:X_k\to X_{k+1}$09. Product systems satisfy corresponding pressure inequalities, with Bowen and packing pressures behaving like Hausdorff and packing dimensions under products (Chen et al., 2 Aug 2025).

The 2025 symbolic paper places nonautonomous symbolic systems within the broader class of strongly uniformly expansive systems. An NDS is strongly uniformly expansive if there exists $T_k:X_k\to X_{k+1}$10 such that for every $T_k:X_k\to X_{k+1}$11 there is $T_k:X_k\to X_{k+1}$12 with

$T_k:X_k\to X_{k+1}$13

for all $T_k:X_k\to X_{k+1}$14 and all $T_k:X_k\to X_{k+1}$15. Nonautonomous symbolic systems are strongly uniformly expansive with expansive constant $T_k:X_k\to X_{k+1}$16. In such systems, generators compute pressure, and if $T_k:X_k\to X_{k+1}$17 is an expansive constant and $T_k:X_k\to X_{k+1}$18, then the fixed-scale pressures already equal the full pressures (Chen et al., 14 Sep 2025).

This expansiveness theory has a universal-coding consequence. Every strongly uniformly expansive nonautonomous system has a symbolic extension: there exist a nonautonomous shift, a closed set $T_k:X_k\to X_{k+1}$19 in its initial symbolic space, and an equicontinuous sequence of surjections intertwining the shift with the original dynamics. If, in addition, each $T_k:X_k\to X_{k+1}$20 has topological dimension zero, then the coding can be made injective; with an additional uniform separation condition for clopen partitions, the embedding is equicontinuous. Since a nonautonomous shift can itself be embedded in an autonomous full shift over a sufficiently large alphabet, a class of zero-dimensional strongly uniformly expansive nonautonomous systems may be embedded in autonomous symbolic systems (Chen et al., 14 Sep 2025).

6. Related constructions, misconceptions, and scope

A common misconception is that positive nonautonomous entropy must reflect persistent asymptotic disorder. Kawan’s survey includes an interval example in which both measure-theoretic and topological entropy equal $T_k:X_k\to X_{k+1}$21 even though the pushed-forward Lebesgue measures converge weakly to $T_k:X_k\to X_{k+1}$22 and every trajectory with initial value in $T_k:X_k\to X_{k+1}$23 converges to $T_k:X_k\to X_{k+1}$24. The stated lesson is that both measure-theoretic and topological entropy can capture transient chaotic behavior. In symbolic terms, this means that exponential finite-window combinatorial growth can coexist with trivial tail behavior (Kawan, 2017).

Another adjacent line of work is indirectly symbolic rather than explicitly shift-theoretic. Štefánková studied surjective interval maps $T_k:X_k\to X_{k+1}$25 converging uniformly to a limit map $T_k:X_k\to X_{k+1}$26 and proved inheritance results for Li–Yorke chaos, distributional chaos, and infinite $T_k:X_k\to X_{k+1}$27-limit sets. Although this is not a paper about subshifts or symbolic spaces, its itinerary lemma, quasi-horseshoes, and binary coding constructions are structurally close to symbolic branch realizations in nonautonomous settings (Štefánková, 2013).

A further terminological boundary concerns the word “symbolic” itself. In the AI and biology literature, van Hateren’s paper on symbol grounding studies representation, reference, and aboutness in embodied and self-reproducing agents. It does not use the term “nonautonomous symbolic systems,” and it is not a contribution to symbolic dynamics in the strict sense. This suggests a strict distinction between two research programs: symbolic dynamics, where symbols are sequence coordinates, cylinders, and coding maps; and symbol-grounding theory, where symbols are semantically interpreted states and the central issue is intrinsic aboutness rather than shift-space complexity (Hateren, 2015).

Taken together, the current mathematical literature presents nonautonomous symbolic systems as the symbolic-dynamics counterpart of general time-dependent dynamics. The foundational entropy theory supplies admissible notions of measure and topological complexity, the subset-pressure theory provides power rules, product rules, and equiconjugacy invariance, and the explicit symbolic formalism yields exact pressure formulas and Bernoulli equilibrium states for strongly bounded variation potentials. At the same time, the general nonautonomous variational principle remains incomplete outside these more structured symbolic settings.