Topological dynamics

Topological dynamics studies continuous maps and their orbits on topological spaces. For a compact metric space $X$ and a continuous map $T:X\to X$, the pair $(X,T)$ is a compact Hausdorff topological dynamical system.

Two continuous maps are topologically conjugate if a homeomorphism $H$ between their state spaces satisfies the displayed intertwining identity. It is conjugate maps with a topology-preserving conjugating map. On compact metric spaces, pulling back a compatible metric through $H$ preserves every Bowen metric, and hence topological entropy.

A continuous map on a nonempty topological space is point-transitive if it has a dense forward orbit. This is one convention for topological transitivity, but on spaces with isolated points it must be distinguished from the open-set intersection definition. On a nonempty compact metric space, the positive-time open-set intersection property implies point transitivity: for each member $U_j$ of a countable basis of a topology, ${\bigcup }_{n\ge 1}{f}^{-n}(U_j)$ is open and dense, and the Baire category theorem supplies a point in their intersection. Conversely, a surjective point-transitive continuous map has every tail of its dense orbit dense, since $f^k(X)=X$ and continuity sends its orbit closure into the closure of that tail. The map $f(1/j)=1/(j+1)$, $f(0)=0$ on $\{0\}\cup \{1/j:j\ge 1\}$ is point-transitive but fails the open-set intersection property.

A subset of the integers is syndetic if it has bounded gaps: some finite length meets the set in every integer interval of that length. On the positive integers, ignoring a finite initial interval gives the same asymptotic notion. Such a set has positive lower density.

A point of a dynamical system is recurrent if some sequence of positive times tending to infinity returns its orbit arbitrarily close to the point. In a compact metric space, this is equivalent to the displayed condition. A periodic point is recurrent, but a recurrent point need not be periodic.

Every continuous self-map of a nonempty compact metric space has a recurrent point. Choose a minimal subsystem. The forward orbit of every point in it is dense, and the same is true of every tail of that orbit; consequently each neighborhood is revisited at arbitrarily large times. A nonempty subsystem is all that is guaranteed: the identity map on a singleton has no larger subsystem.

In a compact metric space with a continuous map $T$, points $x,y$ are proximal ifThis property is independent of the compatible metric. If $T$ is injective and $x\ne y$, arbitrarily small proximal distances must occur at arbitrarily large times, since each finite collection of distances is strictly positive. Proximal points need not be equal or have dense orbits.

If distinct points $x,y$ of a dynamical system on a compact metric space satisfy $d(T^{n}x,T^{n}y)\to 0$ in one compatible metric, no compatible metric can make $T$ an isometry. On a compact space, all compatible metrics give the same asymptotic-pair property by uniform continuity, whereas an isometry preserves the strictly positive distance between distinct points. In a full shift, a constant sequence and a sequence differing at just one coordinate converge to each other under forward shifts, proving this obstruction directly in the product topology.

For a compact metric space with a continuous map $T$, suppose $y$ is a minimal point and $x,y$ are proximal. Every neighborhood $U$ of $y$ admits arbitrarily large $n$ with $T^{n}x,T^{n}y\in U$. First shrink $U$ to an open neighborhood, then choose an open $V$ with $y\in V$ and $\overline{V}\subseteq U$. Minimality and compactness give a finite cover of the orbit closure of $y$ by $T^{-j}V$, $0\le j\le J$. Uniform continuity of these finitely many iterates turns a sufficiently close proximal encounter into a common visit to $U$ after at most $J$ more steps. If the proximal encounters occur only at bounded times, a zero distance at some time makes the two future orbits coincide, and minimality then gives arbitrarily late common visits. Thus injectivity is not needed for this general lemma.

A nonempty topological dynamical system $(Y,T)$ on a compact Hausdorff space is minimal if it has no proper nonempty closed forward-invariant subset. Equivalently, every forward orbit is dense in $Y$, because an orbit closure is closed and forward invariant. Minimality implies $T(Y)=Y$.

A point is minimal when its forward orbit closure is a minimal dynamical system. This does not require the point to be a fixed point. In a finite-alphabet full shift, minimal points are exactly the uniformly recurrent sequences.

A minimal subsystem is a nonempty closed forward-invariant subset on which the restricted dynamical system is a minimal dynamical system. Every continuous map on a nonempty compact Hausdorff space has one: intersections of chains of nonempty closed invariant sets remain nonempty by compactness, so the Zorn lemma gives a minimal member.

The forward orbit closure of $x$ under a continuous map $T$ is the closure of its forward orbit. It is a closed forward-invariant set. If $T$ is invertible, one may instead consider the two-sided orbit closure using $n\in \mathbb{Z}$; the convention should be specified. For a minimal point of an invertible compact system the two closures agree.

Symbolic dynamics studies dynamical systems whose points are sequences of symbols from an alphabet and whose evolution is a shift. A finite alphabet with the discrete topology gives its sequence space a product topology.

A subshift of finite type is a closed subset of a finite-alphabet full shift defined by finitely many forbidden finite words, with the restricted left shift. After replacing symbols by sufficiently long blocks, its constraints can be described by a zero-one symbolic transition matrix. In this presentation, membership means that every consecutive pair follows an allowed transition. The resulting sequence space is a compact metric space and the restricted left shift is a homeomorphism.

A fixed point of the $n$th iteration of a map of a subshift of finite type is determined uniquely by its length-$n$ block with a permitted closing transition. Summing the counts of length-$n$ closed walks over their starting symbols gives the displayed trace. This counts points whose least period divides $n$. If $P_n$ counts points of least period exactly $n$, then $\mathrm{tr}(A^n)={\sum }_{d\mid n}{P}_d$, and Möbius inversion gives $P_n={\sum }_{d\mid n}{\mu }_{\mathrm{M}}(n/d)\mathrm{tr}(A^d)$. The number of distinct least-period-$n$ periodic orbits is $P_n/n$.

The adjacency matrix of a directed graph whose vertices are symbols and whose directed edges are the permitted consecutive pairs. Its entries indicate permission. In comparison, the transition matrix of a Markov chain records transition probabilities.

A finite word over an alphabet is locally admissible for a zero-one symbolic transition matrix if every consecutive pair is permitted. Its length counts symbols, so length $n+1$ corresponds to $n$ transitions. The number with endpoints $i,j$ is $(A^n{)}_{ij}$ by matrix multiplication. Local admissibility does not always imply occurrence in a two-sided subshift of finite type: the matrix $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ permits the word $(1,2)$ but admits no two-sided sequence. If every row and column has a nonzero entry, every locally admissible word extends in both directions.

A two-sided sequence over a finite alphabet is uniformly recurrent if every finite word over an alphabet occurring in it occurs with bounded gaps. More precisely, for each such word some $M$ ensures that every length-$M$ interval contains a complete occurrence. This is equivalent to being a minimal point of the full shift: a finite cover of the orbit closure by preimages of a word's cylinder set bounds its return gaps; conversely, bounded gaps pass to all points of the orbit closure and make every forward orbit dense there. The property concerns finite words, not infinite integer intervals.

The two-sided full shift on a finite alphabet $[k]$ consists of all functions $z:\mathbb{Z}\to [k]$, with the product topology and the left shift. It is a compact metric space. A compatible metric isAgreement on increasingly large finite coordinate sets is equivalent to convergence in this topology. The metric above is compatible but is not invariant under the left shift.

On a two-sided full shift, the left shift is the homeomorphismIts inverse sends $z(j)$ to $z(j-1)$. On a one-sided sequence space, the same forward shift is generally not invertible.