Nonstationary Markov Partitions

• Nonstationary Markov partitions are extensions of classical Markov partitions, adapted to dynamics that vary with time or parameters in both holomorphic and smooth settings.
• They provide robust symbolic models—such as subshifts of finite type and nonstationary edge shifts—that remain invariant even as the underlying maps or algorithms evolve.
• This framework supports applications from persistent graph constructions in rational dynamics to Rauzy fractals in toral automorphisms and time-dependent probabilistic partition processes.

Searching arXiv for the cited works to ground the article in published/preprint sources. Nonstationary Markov partitions are Markov codings adapted to dynamics that vary with time or parameters rather than remaining attached to a single fixed map. In holomorphic dynamics, they appear as families of partitions $P(g)$ whose defining graphs $G(g)$ persist over open regions of parameter space while the associated symbolic system remains constant (Rees, 2013). In smooth and symbolic dynamics on tori, they appear as a sequence $(P_n)_{n\in\mathbb Z}$ attached to a mapping family, with a level-dependent Markov property and a symbolic realization by a nonstationary edge shift (Arnoux et al., 22 Aug 2025). A distinct probabilistic usage concerns Markov processes whose states are set partitions; there, nonstationarity refers either to starting away from the stationary law or, as a natural extension of the formalism, to time-dependent directing measures on random maps (Crane, 2011, Crane, 2015).

1. Definitions and conceptual scope

The stationary starting point is the classical Markov partition for a single map. For a rational map $f:\hat{\mathbb C}\to\hat{\mathbb C}$ of degree at least $2$, a Markov partition is a finite collection $P=\{P_1,\dots,P_n\}$ of closed subsets of $\hat{\mathbb C}$ with disjoint interiors whose union is the whole sphere, and such that each $P_i$ is a union of connected components of $f^{-1}(P_j)$ for varying $j$. This yields a transition matrix

$G(g)$0

and hence a subshift of finite type $G(g)$1 together with a semi-conjugacy $G(g)$2 satisfying $G(g)$3 (Rees, 2013).

The genuinely nonstationary formulation replaces a single map by a mapping family

$G(g)$4

A sequence $G(g)$5 of topological partitions of $G(g)$6 is a nonstationary Markov partition when one-step admissibility along a finite window implies nonempty total intersection over that window. Concretely, for each $G(g)$7, $G(g)$8, and index sequence $G(g)$9,

$(P_n)_{n\in\mathbb Z}$0

implies

$(P_n)_{n\in\mathbb Z}$1

The generating property is defined by requiring that the intersection of all forward and backward iterates of a bi-infinite itinerary contain at most one point (Arnoux et al., 22 Aug 2025).

A practical criterion for this nonstationary Markov property is Property M: each atom $(P_n)_{n\in\mathbb Z}$2 carries transverse partitions $(P_n)_{n\in\mathbb Z}$3 and $(P_n)_{n\in\mathbb Z}$4 such that, whenever $(P_n)_{n\in\mathbb Z}$5,

$(P_n)_{n\in\mathbb Z}$6

If $(P_n)_{n\in\mathbb Z}$7 has Property M, then it is a nonstationary Markov partition (Arnoux et al., 22 Aug 2025).

2. Persistent Markov partitions for rational maps

In rational dynamics, the nonstationary aspect is parameter dependence. Rees studies rational maps for which every critical point lies in the Fatou set, the closure of each Fatou component is a closed topological disk, closures of distinct components are disjoint, the union $(P_n)_{n\in\mathbb Z}$8 of periodic Fatou components is a finite union of closed disks, and there is a finite forward-invariant set $(P_n)_{n\in\mathbb Z}$9 containing all parabolic points. Given an initial graph $f:\hat{\mathbb C}\to\hat{\mathbb C}$0 with finitely many trivalent vertices and edges, with topological-disk complementary components and at most one component of $f:\hat{\mathbb C}\to\hat{\mathbb C}$1 in each complementary disk, Theorem 1.1 produces a graph $f:\hat{\mathbb C}\to\hat{\mathbb C}$2 ambient isotopic to $f:\hat{\mathbb C}\to\hat{\mathbb C}$3 and an integer $f:\hat{\mathbb C}\to\hat{\mathbb C}$4 such that

$f:\hat{\mathbb C}\to\hat{\mathbb C}$5

Moreover, $f:\hat{\mathbb C}\to\hat{\mathbb C}$6 can be chosen arbitrarily close in Hausdorff metric to $f:\hat{\mathbb C}\to\hat{\mathbb C}$7, and its closed loops approximate closed loops of that union that bound disks of diameter bounded below. From this one obtains a finite connected graph $f:\hat{\mathbb C}\to\hat{\mathbb C}$8 with

$f:\hat{\mathbb C}\to\hat{\mathbb C}$9

and the complementary components of $2$0 form a Markov partition. Corollary 1.2 further arranges that $2$1 avoids the closures of periodic Fatou components and that each partition element contains at most one periodic Fatou component (Rees, 2013).

The parameter-persistent statement is Theorem 2.1. Let $2$2 be the critical value set of $2$3, and suppose that a graph $2$4 satisfies four conditions: $2$5; each edge $2$6 of $2$7 is expanded in the sense that $2$8 contains more than a single edge of $2$9 for all large $P=\{P_1,\dots,P_n\}$0; $P=\{P_1,\dots,P_n\}$1 separates the points of $P=\{P_1,\dots,P_n\}$2; and for some $P=\{P_1,\dots,P_n\}$3, the set $P=\{P_1,\dots,P_n\}$4 is separated from $P=\{P_1,\dots,P_n\}$5 by $P=\{P_1,\dots,P_n\}$6. Then every rational map $P=\{P_1,\dots,P_n\}$7 sufficiently close to $P=\{P_1,\dots,P_n\}$8 in the uniform topology admits a graph $P=\{P_1,\dots,P_n\}$9 such that $\hat{\mathbb C}$0, $\hat{\mathbb C}$1 is ambient isotopic to $\hat{\mathbb C}$2, and the isotopy varies continuously with $\hat{\mathbb C}$3. Expansion of $\hat{\mathbb C}$4 near $\hat{\mathbb C}$5 for large $\hat{\mathbb C}$6 persists for nearby maps, so the complements

$\hat{\mathbb C}$7

form Markov partitions with constant combinatorics across the neighborhood. This is the precise content of a persistent Markov partition: a partition that exists on an open region of parameter space, not only on a single map or a single hyperbolic component (Rees, 2013).

3. Geometric and analytic machinery in the rational case

The construction rests on expansion and pullback control near the invariant graph. For a suitable neighborhood $\hat{\mathbb C}$8 of $\hat{\mathbb C}$9, some iterate $P_i$0 is expanding in the spherical metric: $P_i$1 with $P_i$2. If $P_i$3 is a compact set disjoint from the postcritical set, then diameters of pullbacks by local inverse branches $P_i$4 of $P_i$5 satisfy

$P_i$6

This is the Brolin argument used to control the geometry of iterated preimages of the initial graph (Rees, 2013).

A central combinatorial device is the nested sequence of arcs in preimages of the graph. Such a sequence $P_i$7 has endpoints on a small arc $P_i$8 of diameter at most $P_i$9, interiors disjoint from $f^{-1}(P_j)$0, each $f^{-1}(P_j)$1 contained in the topological disk bounded by $f^{-1}(P_j)$2, and $f^{-1}(P_j)$3. Rees proves that there is only a finite combinatorial set $f^{-1}(P_j)$4 of possible local patterns for such nested sequences, and that their diameters decay exponentially: $f^{-1}(P_j)$5 for some $f^{-1}(P_j)$6 and $f^{-1}(P_j)$7. This confines accumulation to a controlled set, a finite collection of small disks with locally connected boundary (Rees, 2013).

The finite graph $f^{-1}(P_j)$8 is obtained by approximation, not by asserting finiteness of the full invariant pullback set. Starting from

$f^{-1}(P_j)$9

which need not itself be a finite graph, the argument uses Hausdorff convergence, expansion estimates, and the $j$0-lemma for holomorphic motions to show that there exists a finite graph $j$1 arbitrarily close to $j$2 in Hausdorff metric with $j$3. The closed loops of $j$4 approximate those of $j$5 that bound large disks. The approximation step is explicitly described as robust, whereas earlier attempts to prove that $j$6 itself is finite are false in general (Rees, 2013).

Parameter dependence is handled analytically. Each partition piece $j$7 is conformally mapped to the unit disk by a Riemann map $j$8; on boundaries, $j$9 restricts to Blaschke products on the unit circle. Edges of $G(g)$00 become intervals on the circle, and on each interval one obtains real-analytic maps $G(g)$01 and $G(g)$02 describing forward dynamics. Under boundedness assumptions on moduli of quadrilaterals and annuli, these maps are uniformly expanding for large iterates and are quasi-symmetrically conjugate with norm bounded independently of $G(g)$03. The resulting quasi-conformal uniformization produces a canonically defined graph $G(g)$04 varying continuously, indeed quasi-conformally, with the parameter (Rees, 2013).

4. Persistence regions, symbolic constancy, and boundary degeneration

Let $G(g)$05 be a connected component of parameter space and $G(g)$06 a base map. Rees defines $G(g)$07 as the set of maps $G(g)$08 for which there is a graph $G(g)$09 satisfying the required invariance and isotopy conditions, and $G(g)$10 as the locus where the dynamical system on the graph, including the forward orbit of critical values, is topologically conjugate up to isotopy to that for the base map. By Theorem 2.1 together with the parameter-variation arguments of Sections 2.3–2.7, $G(g)$11 is open. If $G(g)$12 is parabolic, or more generally non-hyperbolic with critical orbits attracted to cycles, the resulting neighborhood necessarily intersects more than one hyperbolic component, so persistence is not confined to a single hyperbolic region (Rees, 2013).

The decisive boundary statement is Theorem 2.2. On a maximal connected open set $G(g)$13 where there exists an isotopically varying graph $G(g)$14 that separates critical values and where the critical values are separated from $G(g)$15 by a fixed combinatorial depth $G(g)$16, any subset $G(g)$17 whose closure intersects $G(g)$18 must satisfy that $G(g)$19 is unbounded on $G(g)$20. Approaching the boundary of the persistence region therefore forces the number of preimage layers needed to separate the critical values from the graph to go to infinity. This gives a precise combinatorial description of graph degeneration at the boundary (Rees, 2013).

Inside $G(g)$21, the graph combinatorics and hence the transition matrix

$G(g)$22

are constant up to relabelling. The symbolic system $G(g)$23 is therefore common to all parameters in the persistence region, with semi-conjugacies

$G(g)$24

Along any continuous path $G(g)$25 in the persistence region, one has a Markov partition $G(g)$26 for each $G(g)$27, while the symbolic space $G(g)$28 and the shift $G(g)$29 are independent of $G(g)$30. The coding maps $G(g)$31 vary continuously, and on graph edges they vary real-analytically. This realizes a nonstationary family with a stationary symbolic skeleton (Rees, 2013).

5. Anosov mapping families, continued fractions, and Rauzy-box partitions

A second, explicitly time-dependent theory appears for toral automorphisms. Given a bi-infinite sequence $G(g)$32 of unimodular integer matrices, one obtains the $G(g)$33-adic mapping family

$G(g)$34

where each $G(g)$35 is a copy of $G(g)$36. The sequences of matrices of interest arise from the natural extensions of multidimensional continued fraction algorithms: a $G(g)$37-dimensional algorithm is a triple $G(g)$38 with $G(g)$39, $G(g)$40, and

$G(g)$41

Passing to the natural extension yields bi-infinite sequences $G(g)$42 with determinant $G(g)$43 (Arnoux et al., 22 Aug 2025).

Hyperbolicity is expressed by a Pisot-type condition. If $G(g)$44 are the singular values and $G(g)$45, then the local Pisot condition in the future is

$G(g)$46

with an analogous past condition. Under the additional mild growth assumption

$G(g)$47

Lemma 2.14 and Proposition 2.15 show that this is equivalent to exponential convergence of the columns of $G(g)$48 to a single direction $G(g)$49, and to exponential contraction of $G(g)$50 on some hyperplane $G(g)$51. For two-sided primitive sequences satisfying the two-sided local Pisot condition and the growth condition in both directions, Theorem 2.21 states that the associated mapping family is eventually Anosov for the splitting

$G(g)$52

where $G(g)$53 and $G(g)$54 are generalized right and left eigenvectors (Arnoux et al., 22 Aug 2025).

To construct partition atoms, the paper superimposes a combinatorial structure through substitutions and $G(g)$55-adic dynamical systems. For a primitive sequence of substitutions $G(g)$56 with incidence matrices $G(g)$57, one defines Rauzy fractals $G(g)$58 and then Rauzy boxes

$G(g)$59

Under the tiling condition, the full boxes $G(g)$60 tile $G(g)$61 modulo $G(g)$62, so their images define topological partitions of $G(g)$63. The set equation for Rauzy fractals translates into a decomposition of Rauzy boxes compatible with the matrices $G(g)$64, and this verifies Property M. Theorem 4.13 then gives a generating nonstationary Markov partition whose atoms are these Rauzy boxes, up to translation, and whose symbolic model is a nonstationary edge shift (Arnoux et al., 22 Aug 2025).

The guiding examples are the unordered, ordered, and multiplicative Brun algorithms. In dimensions $G(g)$65, Proposition 3.26 states that each of these algorithms satisfies the Pisot condition, so almost every orbit yields an Anosov mapping family with codimension-one hyperbolicity. Corollary G further states that in dimension $G(g)$66, for almost every pair of complementary subspaces $G(g)$67, there is a linear Anosov mapping family defined by the Brun algorithm with stable foliation $G(g)$68, unstable foliation $G(g)$69, a generating Markov partition, and a symbolic model as a nonstationary edge shift; an analogous statement holds in dimension $G(g)$70 when $G(g)$71 (Arnoux et al., 22 Aug 2025).

6. Partition-valued Markov processes and probabilistic nonstationarity

A different but mathematically precise use of the phrase concerns Markov processes on spaces of set partitions. In Crane’s framework, the state space is

$G(g)$72

the partitions of $G(g)$73 with at most $G(g)$74 blocks. The construction is driven by Kingman’s paintbox on ranked-mass partitions $G(g)$75 with $G(g)$76, restricted to $G(g)$77, and by a Poisson point process on

$G(g)$78

with intensity $G(g)$79, or $G(g)$80. At each atom $G(g)$81, a matrix construction fragments each current block $G(g)$82 according to $G(g)$83, applies independent uniform permutations of $G(g)$84 to the columns, and then forms column totals

$G(g)$85

This yields an exchangeable, consistent Markov process $G(g)$86 on $G(g)$87, the $G(g)$88-Markov process (Crane, 2011).

For the finite restrictions to $G(g)$89, the process is irreducible whenever

$G(g)$90

and aperiodic because $G(g)$91 for every state. Hence there exists a unique exchangeable stationary probability measure $G(g)$92 on $G(g)$93 whose restrictions are the unique stationary laws of the finite chains. In this setting, nonstationary behavior means that the initial law $G(g)$94 is not equal to $G(g)$95. The process is time-homogeneous; nonstationarity refers to the initial distribution, not to time-dependent transition rules (Crane, 2011).

Lipschitz partition processes generalize this by placing the Poisson construction on the space $G(g)$96 of $G(g)$97-Lipschitz maps on labeled $G(g)$98-partitions, with finite-level rate condition

$G(g)$99

The resulting process is Feller and càdlàg, and its finite restrictions form a compatible family of finite-state Markov chains. In the exchangeable case, the directing measure is supported on strongly Lipschitz maps, characterized by the overlap condition

$(P_n)_{n\in\mathbb Z}$00

These maps are in one-to-one correspondence with set-valued partition operators, namely $(P_n)_{n\in\mathbb Z}$01 matrices $(P_n)_{n\in\mathbb Z}$02 acting by

$(P_n)_{n\in\mathbb Z}$03

The associated asymptotic-frequency process is driven by column-stochastic matrices $(P_n)_{n\in\mathbb Z}$04 and is again Feller. The paper itself remains time-homogeneous, but it explicitly notes that allowing a time-dependent intensity $(P_n)_{n\in\mathbb Z}$05 would yield a time-inhomogeneous Markov process on $(P_n)_{n\in\mathbb Z}$06, provided analogues of the finite-rate conditions hold uniformly in $(P_n)_{n\in\mathbb Z}$07 (Crane, 2015).

7. Comparative perspective and principal open directions

The cited literature therefore uses nonstationary Markov partitions in two distinct but structurally related ways. In dynamical systems, the phrase refers to Markov codings for varying maps: in the rational case, a graph and its complementary regions move through parameter space while the transition matrix stays fixed on a persistence region; in the toral case, the atoms themselves vary with the time index $(P_n)_{n\in\mathbb Z}$08 and are tied to an Anosov mapping family by Property M. In the probabilistic literature, by contrast, the state of the process is itself a partition, and nonstationarity concerns either the choice of initial law or, in a natural extension of the Lipschitz-map formalism, time-dependent transition mechanisms (Rees, 2013, Arnoux et al., 22 Aug 2025, Crane, 2011, Crane, 2015).

In the dynamical setting, a unifying theme is the coexistence of geometric variability and symbolic rigidity. Rees’s persistence regions exhibit graphs $(P_n)_{n\in\mathbb Z}$09 and partition elements $(P_n)_{n\in\mathbb Z}$10 that deform continuously while a single subshift of finite type remains valid; loss of persistence is detected by the blow-up of the combinatorial depth $(P_n)_{n\in\mathbb Z}$11. For toral mapping families, Rauzy boxes provide atoms that are explicit geometric realizations of $(P_n)_{n\in\mathbb Z}$12-adic systems by suspensions of $(P_n)_{n\in\mathbb Z}$13-adic Rauzy fractals, and the resulting symbolic model is a nonstationary edge shift (Rees, 2013, Arnoux et al., 22 Aug 2025).

The principal open directions stated in the cited works are likewise different in the two traditions. For multidimensional continued fractions, open problems include establishing Pisot conditions in higher dimensions, proving tiling properties and pure discrete spectrum for general $(P_n)_{n\in\mathbb Z}$14-adic systems satisfying Pisot conditions, characterizing substitutive realizations for given algorithms, and understanding the interaction between geometry and combinatorics in greater generality (Arnoux et al., 22 Aug 2025). For Lipschitz partition processes, the undeveloped extension to time-dependent directing measures raises questions about preserving consistency and the Feller property, characterizing time-inhomogeneous exchangeable processes, and extending the framework beyond a bounded number of blocks (Crane, 2015). A plausible implication is that the common conceptual core is not a fixed partition but a coherent symbolic or probabilistic structure that remains well defined while the underlying dynamics vary.