Nonautonomous Iterated Function Systems

• Nonautonomous Iterated Function Systems are generalizations of classical IFS where the contraction maps vary with time, enabling dynamic symbolic coding and diverse geometric configurations.
• They extend dimension theory using singular-value functions, pressure formulations, and multifractal spectra to handle overlaps and level-dependent variability.
• The framework supports both affine and conformal settings, providing rigorous results on Hausdorff and box-counting dimensions under various separation and transversality conditions.

Nonautonomous iterated function systems are generalizations of classical iterated function systems in which the contractions applied at each step are allowed to vary with time. In geometric formulations one fixes a compact set $J\subset\mathbb R^d$ with nonempty interior and, at level $k$, chooses $n_k$ contractive maps $S_{k,i}:J\to J$; the attractor is then

$E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$

In conformal settings the corresponding objects are non-autonomous conformal iterated function systems, while in topological dynamics one also studies sequences of finite families of continuous maps on a compact metric space. Across these variants, the subject is organized around symbolic coding, pressure, dimension theory, overlaps, and the effect of time dependence on the relation between Hausdorff and box-type dimensions (Gu et al., 2023, Rempe-Gillen et al., 2012, Ju et al., 30 Jul 2025).

1. Formal setups and symbolic organization

The literature uses several closely related formalizations. In the affine-geometric framework, one starts with a sequence $\{n_k\}_{k\ge1}$ of integers $\ge2$, the word spaces

$$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$

and the infinite coding space $\Sigma^\infty$. For $u=u_1\cdots u_k\in\Sigma^*$, one writes $k$0, $k$1, and defines cylinders $k$2. At level $k$3, the non-autonomous IFS consists of contractive maps $k$4 with

$k$5

and no separation assumptions are imposed (Gu et al., 2023).

A broader dynamical formulation considers a compact metric space $k$6 and a sequence of finite families of continuous maps,

$k$7

with words $k$8 and compositions

$k$9

This is the setting in which topological pressure and factor maps are studied (Ju et al., 30 Jul 2025).

In the conformal theory, one assumes that $n_k$0 is compact with $n_k$1, that each level $n_k$2 carries a family

$n_k$3

and that the maps satisfy an open-set condition, uniform contraction, and bounded distortion. A related recent formulation uses a fixed compact $n_k$4 with nonempty interior, level families $n_k$5, and a uniform Lipschitz bound $n_k$6 (Rempe-Gillen et al., 2012, Miao et al., 28 Aug 2025).

A distinct but compatible point of view replaces the full shift by an arbitrary subshift $n_k$7. One then studies a generalized IFS

$n_k$8

with admissible words drawn from the language $n_k$9, and for each driving sequence $S_{k,i}:J\to J$0 one obtains a non-autonomous system

$S_{k,i}:J\to J$1

This shows that the term “non-autonomous” includes both level-dependent families and dynamics along a single admissible coding sequence (Dastjerdi et al., 2022).

Setting	Basic data	Central object
Geometric NIFS	$S_{k,i}:J\to J$2	$S_{k,i}:J\to J$3
NCIFS	conformal contractions with distortion control	limit set $S_{k,i}:J\to J$4
NAIFS on metric spaces	$S_{k,i}:J\to J$5	pressure and factor-map theory
Arbitrary-shift IFS	$S_{k,i}:J\to J$6 plus a subshift $S_{k,i}:J\to J$7	non-autonomous subsystem $S_{k,i}:J\to J$8

2. Attractors, compositions, and affine specialization

In the geometric construction, for a word $S_{k,i}:J\to J$9 one defines a composed map

$E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$0

where each $E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$1 is a translation chosen to place the image inside $E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$2. The level-$E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$3 basic sets are $E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$4, and the NIFS attractor is

$E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$5

Because no separation assumption is built into the definition, overlap is part of the basic model rather than an exceptional phenomenon (Gu et al., 2023).

The affine subclass is obtained by restricting to maps of the form

$E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$6

where $E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$7 is a non-singular linear contraction, $E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$8, and $E=\bigcap_{k=1}^\infty \bigcup_{u\in\Sigma^k} J_u.$9. Writing $\{n_k\}_{k\ge1}$0, one has

$\{n_k\}_{k\ge1}$1

These non-autonomous affine sets are the main setting for singular-value methods and for the separation between different dimension exponents (Gu et al., 2023).

In the conformal theory, the attractor has an equivalent coding description. If $\{n_k\}_{k\ge1}$2, then

$\{n_k\}_{k\ge1}$3

and

$\{n_k\}_{k\ge1}$4

Each infinite sequence $\{n_k\}_{k\ge1}$5 determines a unique point

$\{n_k\}_{k\ge1}$6

so the attractor is the image of symbolic space under $\{n_k\}_{k\ge1}$7 (Miao et al., 28 Aug 2025).

The absence of separation can change the geometric character of the limit set drastically. Non-autonomous Moran-type examples show that without separation assumptions $\{n_k\}_{k\ge1}$8 can be finite, countable or of positive Lebesgue measure. This precludes any universal dimension formula that ignores overlaps or the levelwise variability of the system (Gu et al., 2023).

3. Critical exponents, pressure, and dimension formulae

For non-autonomous affine sets, dimension theory is organized by the singular-value function. If a linear map $\{n_k\}_{k\ge1}$9 has singular values

$\ge2$0

then for $\ge2$1 one defines

$\ge2$2

and for $\ge2$3,

$\ge2$4

The function $\ge2$5 is continuous and sub-multiplicative: $\ge2$6 Using cut-sets

$\ge2$7

one defines the box-critical exponent

$\ge2$8

and the net-measure exponent $\ge2$9 through the outer net measure

$$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$0

with

$$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$1

For any non-autonomous affine attractor $$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$2,

$$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$3

Unlike self-affine fractals where $$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$4, one always has $$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$5, and the inequality may strictly hold (Gu et al., 2023).

Under additional hypotheses, these upper bounds become exact. If the system satisfies the open projection condition (OPC), then

$$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$6

Under the same OPC and a uniform lower bound on projected measures,

$$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$7

one also gets

$$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$8

Accordingly, the upper box-counting dimension and Hausdorff dimension may equal $$\Sigma^k=\{u_1\cdots u_k:1\le u_j\le n_j\},\qquad \Sigma^*=\bigcup_k\Sigma^k,$$9 and $\Sigma^\infty$0, respectively (Gu et al., 2023).

The conformal theory is formulated through partition sums and pressure. For words $\Sigma^\infty$1,

$\Sigma^\infty$2

and the lower pressure is

$\Sigma^\infty$3

The Bowen dimension is

$\Sigma^\infty$4

Always $\Sigma^\infty$5. Bowen’s formula, namely $\Sigma^\infty$6, holds for finite-alphabet systems under the sub-exponential growth condition

$\Sigma^\infty$7

More generally, if $\Sigma^\infty$8 and either the largest contraction tends to $\Sigma^\infty$9, or $u=u_1\cdots u_k\in\Sigma^*$0, or $u=u_1\cdots u_k\in\Sigma^*$1, then Bowen’s formula holds as well. In the regular exponential-growth case

$u=u_1\cdots u_k\in\Sigma^*$2

one obtains $u=u_1\cdots u_k\in\Sigma^*$3 and hence $u=u_1\cdots u_k\in\Sigma^*$4 (Rempe-Gillen et al., 2012).

A recent extension introduces intermediate dimension spectra for non-autonomous conformal sets. For $u=u_1\cdots u_k\in\Sigma^*$5, one defines upper and lower pressures $u=u_1\cdots u_k\in\Sigma^*$6 and $u=u_1\cdots u_k\in\Sigma^*$7, their critical values

$u=u_1\cdots u_k\in\Sigma^*$8

$u=u_1\cdots u_k\in\Sigma^*$9

and under OSC and the mild “no-extreme-ratio” hypothesis $k$00 one has

$k$01

Setting $k$02 gives the Hausdorff dimension, while setting $k$03 gives the box-counting dimension and also packing dimension: $k$04 This provides a pressure-zero formalism interpolating between Hausdorff and box-counting scales (Miao et al., 28 Aug 2025).

4. Overlaps, transversality, and almost-sure dimension

One of the central technical distinctions in the subject is between deterministic separation hypotheses and almost-sure statements that persist in the presence of overlap. In the affine theory, if the translations $k$05 are chosen i.i.d. from a bounded region with an absolutely continuous law, then almost surely the random attractor $k$06 satisfies

$k$07

and

$k$08

These results use a Frostman-type measure on symbolic space and a potential integral estimate proved by conditioning the i.i.d. translations (Gu et al., 2023).

The conformal overlap theory reaches an analogous conclusion through transversality. A transversal family of non-autonomous conformal IFS is a parameterized family $k$09 satisfying conformality, uniform contraction, bounded distortion, distortion continuity, continuity of addresses, and a transversality bound

$k$10

For such a family, if $k$11 denotes the limit set and $k$12 the Bowen dimension, then for Lebesgue-almost every $k$13,

$k$14

Moreover, for almost every $k$15 such that $k$16,

$k$17

This gives almost-sure exact dimension and positive-volume statements without the open set condition (Nakajima, 2023).

A concrete example makes the role of transversality explicit. In $k$18, for each $k$19 define

$k$20

For a suitable parameter domain $k$21, the resulting family satisfies the transversality condition but does not satisfy the open set condition for any $k$22. In this example,

$k$23

so for almost every $k$24,

$k$25

This shows that failure of OSC does not by itself prevent an almost-sure Bowen-type dimension formula (Nakajima, 2023).

The comparison with separation-based results is instructive. In some parts of the theory, such as the OPC framework for affine systems or OSC/SSC frameworks for conformal and similar systems, separation is used to establish deterministic exact formulae. In the transversality and random-translation settings, overlap is retained, but exact formulae are recovered for almost every parameter or almost every translation realization. A plausible implication is that non-autonomous dimension theory is governed as much by the structure of overlap as by the contraction data themselves.

5. Measures and generalized $k$26-dimensions

Beyond set dimensions, nonautonomous fractals support a measure-theoretic dimension theory based on generalized $k$27-dimensions. For a Borel probability measure $k$28 on $k$29, one defines for $k$30

$k$31

and for $k$32,

$k$33

Equivalent ball-integral formulations are also available (Gu et al., 2024).

For nonautonomous similar attractors, where each $k$34 is a similarity of ratio $k$35, and $k$36 is the projection of levelwise Bernoulli weights $k$37, the strong separation condition yields an exact formula. If

$k$38

then for all $k$39,

$k$40

where $k$41. Under mild regularity, $k$42, so $k$43. In the level-independent case $k$44 and $k$45, this recovers the familiar self-similar relation

$k$46

The thermodynamic quantity controlling the proof is

$k$47

(Gu et al., 2024).

For nonautonomous affine sets, with maps

$k$48

one replaces similarity ratios by the singular-value function

$k$49

For $k$50, define

$k$51

$k$52

Then, in full generality,

$k$53

The proof uses the fact that each basic set $k$54 is contained in a parallelepiped whose side lengths are the singular values of $k$55 (Gu et al., 2024).

Two special affine models admit exact formulas for $k$56. In the almost self-affine case, where the linear parts are level-independent and the translations are i.i.d. random vectors with absolutely continuous density on a bounded region, one has almost surely

$k$57

In the finite-translation nonautonomous case, for Lebesgue-a.e. translation choice $k$58,

$k$59

and under additional nondegeneracy this extends to $k$60 (Gu et al., 2024).

These results place measures on nonautonomous fractals within the same level-by-level thermodynamic framework that governs set dimensions, but they also show that the relevant pressure objects depend on the order $k$61, on the coding measure, and on whether the geometry is similar or affine.

6. Dynamical variants, factor maps, and sharpness phenomena

The thermodynamic formalism for NAIFS on compact metric spaces includes a factor-map inequality for topological pressure. Given equicontinuous NAIFSs $k$62 and $k$63 with the same index sets and a semiconjugacy $k$64, if $k$65 is continuous, then

$k$66

where $k$67 is the topological sup-entropy of the fiber. If $k$68 is a conjugacy, then

$k$69

When each $k$70 has exactly one map, the formalism reduces to a non-autonomous system $k$71; when all $k$72 are the same finite family, one recovers free semigroup actions (Ju et al., 30 Jul 2025).

The arbitrary-shift approach clarifies how non-autonomous behavior can be encoded by a single admissible sequence. For a generalized IFS $k$73, one distinguishes transitivity of the IFS as a whole from transitivity along a sequence $k$74. Even if $k$75 is topologically transitive, it need not admit any $k$76 such that $k$77 is transitive. Under additional hypotheses—$k$78 compact metric, $k$79 an irreducible sofic shift, the maps $k$80 surjective and semi-open, and the IFS topologically transitive—there exists a point $k$81 which is forward-transitive for the shift and a point $k$82 such that

$k$83

For the set

$k$84

several behaviors are possible: $k$85 may be empty; $k$86 may coincide with all of $k$87; $k$88 may be dense, uncountable, yet not residual; and under a mixing SFT with a unique measure of maximal entropy, $k$89 has full measure even though it need not be residual (Dastjerdi et al., 2022).

Sharpness results show that the non-autonomous theory cannot be reduced to a naïve extension of autonomous formulas. In the affine setting one can construct a $k$90-dimensional non-autonomous affine system where $k$91 and indeed

$k$92

strictly. In the autonomous case, by contrast, one has a unique affinity dimension $k$93 characterized by

$k$94

and then

$k$95

This recovers Falconer’s almost-sure Hausdorff-dimension formula (Gu et al., 2023).

The conformal theory yields an analogous sharpness statement for growth conditions. For any $k$96 and $k$97, one can construct a perfectly balanced example with

$k$98

but

$k$99

showing that the sub-exponential hypothesis in Bowen’s formula is sharp. In the infinite-alphabet case, continued-fraction constructions produce systems with $n_k$00 but $n_k$01, illustrating that infinite non-autonomous systems require additional tail-control hypotheses (Rempe-Gillen et al., 2012).

Taken together, these results establish nonautonomous iterated function systems as a theory in which symbolic combinatorics, pressure, overlap geometry, and levelwise variability interact more delicately than in the autonomous case. A plausible implication is that the non-autonomous regime is best understood not as a perturbation of ordinary IFS theory, but as a framework in which multiple critical exponents, multiple pressures, and multiple dynamical codings coexist and need not collapse to a single invariant.