Frequent Hypercyclicity Criterion

• Frequent Hypercyclicity Criterion is a sufficient condition in linear dynamics that guarantees an operator's orbit hits every non-empty open set with positive lower density.
• It relies on a dense subset and an associated mapping that ensures the unconditional convergence of series, enhancing the classical hypercyclicity framework with controlled recurrence.
• This criterion underpins the study of hypercyclic subspaces, invariant strongly mixing measures, and quantitative recurrence scales in various operator and sequence settings.

The Frequent Hypercyclicity Criterion is a sufficient criterion in linear dynamics ensuring that a continuous linear operator, or more generally a sequence of operators, admits an orbit visiting every non-empty open set with positive lower density. In its classical operator form, due to Bayart–Grivaux and used throughout later work, it requires a dense subset $X_0\subset X$ and a map $S:X_0\to X_0$ such that for every $x\in X_0$, the series $\sum_{n=0}^\infty T^n x$ and $\sum_{n=0}^\infty S^n x$ converge unconditionally in $X$, and $TSx=x$ (Ernst et al., 2016, Agneessens, 2022). The criterion is stronger than the classical Hypercyclicity Criterion, but precisely this extra strength makes it a key analytic tool for proving frequent hypercyclicity, analyzing hypercyclic subspaces, constructing invariant strongly mixing measures, and quantifying recurrence beyond natural density (Menet, 2013).

1. Classical and sequence formulations

For an operator $T$ on a separable Fréchet space $X$, frequent hypercyclicity means that there exists $x\in X$ such that for every non-empty open set $S:X_0\to X_0$0, the return set

$S:X_0\to X_0$1

has positive lower density,

$S:X_0\to X_0$2

The classical Frequent Hypercyclicity Criterion states that this follows if there are a dense set $S:X_0\to X_0$3 and a map $S:X_0\to X_0$4 with

$S:X_0\to X_0$5

unconditionally convergent for every $S:X_0\to X_0$6, together with $S:X_0\to X_0$7 (Agneessens, 2022).

A broader formulation treats sequences of operators $S:X_0\to X_0$8, where $S:X_0\to X_0$9 is a Fréchet space and $x\in X_0$0 is a separable Fréchet space. In that setting, $x\in X_0$1 satisfies the Frequent Hypercyclicity Criterion if there exist a dense subset $x\in X_0$2 and maps $x\in X_0$3 such that for each $x\in X_0$4,

$x\in X_0$5

converge unconditionally in $x\in X_0$6, uniformly in $x\in X_0$7,

$x\in X_0$8

converges unconditionally in $x\in X_0$9, and

$\sum_{n=0}^\infty T^n x$0

Here unconditional convergence uniformly in $\sum_{n=0}^\infty T^n x$1 means that for every $\sum_{n=0}^\infty T^n x$2 there exists $\sum_{n=0}^\infty T^n x$3 such that for all $\sum_{n=0}^\infty T^n x$4 in the relevant index set and all finite $\sum_{n=0}^\infty T^n x$5,

$\sum_{n=0}^\infty T^n x$6

where $\sum_{n=0}^\infty T^n x$7 denotes the corresponding series terms (Menet, 2013).

This sequence form is the natural framework for the Bonilla–Grosse-Erdmann frequent universality machinery, and the single-operator criterion is recovered by taking $\sum_{n=0}^\infty T^n x$8.

2. Relation to the Hypercyclicity Criterion and universality

The classical Hypercyclicity Criterion requires dense sets $\sum_{n=0}^\infty T^n x$9, $\sum_{n=0}^\infty S^n x$0, an increasing sequence $\sum_{n=0}^\infty S^n x$1, and maps $\sum_{n=0}^\infty S^n x$2 such that

$\sum_{n=0}^\infty S^n x$3

Under these hypotheses, $\sum_{n=0}^\infty S^n x$4 is weakly mixing and hence hypercyclic; for operators, the Hypercyclicity Criterion is equivalent to weak mixing and to hereditary hypercyclicity (Menet, 2013).

The Frequent Hypercyclicity Criterion is stronger in two distinct senses. First, it replaces convergence-to-zero requirements by unconditional convergence of whole series, uniformly in the relevant parameter. Second, it upgrades qualitative recurrence to quantitative recurrence: the orbit does not merely meet every non-empty open set infinitely often, but does so with positive lower density. In the sequence formulation, the criterion implies the Hypercyclicity Criterion along the full sequence $\sum_{n=0}^\infty S^n x$5 (Menet, 2013).

Conceptually, the ordinary Hypercyclicity Criterion provides a mechanism for infinitely many returns, whereas the Frequent Hypercyclicity Criterion provides enough combinatorial and summability structure to force returns with controlled frequency. This distinction becomes decisive in later developments: common frequent hypercyclicity for families of operators, weighted-density refinements, and the construction of invariant strongly mixing measures all rely on structure absent from the classical criterion (Charpentier et al., 2020).

3. Orbit structure, hypercyclic subspaces, and hereditary forms

One of the most precise structural uses of the Frequent Hypercyclicity Criterion appears in Menet’s characterization of hypercyclic subspaces. If $\sum_{n=0}^\infty S^n x$6 is an infinite-dimensional separable Fréchet space with a continuous norm and $\sum_{n=0}^\infty S^n x$7 satisfies the Frequent Hypercyclicity Criterion, then the following are equivalent: $\sum_{n=0}^\infty S^n x$8 possesses a hypercyclic subspace; there exists an infinite-dimensional closed subspace $\sum_{n=0}^\infty S^n x$9 such that for every continuous seminorm $X$0 on $X$1,

$X$2

and there exist an infinite-dimensional closed subspace $X$3 and an increasing sequence $X$4 such that for every continuous seminorm $X$5,

$X$6

In the same paper, an analogous characterization is obtained for sequences $X$7 satisfying the Hypercyclicity Criterion along a subsequence, now phrased in terms of hereditarily hypercyclic subspaces along that subsequence (Menet, 2013).

The significance of these equivalences is that, under FHC, the existence of hypercyclic subspaces becomes a purely orbit-theoretic question of controlling seminorm growth on an infinite-dimensional subspace. The resulting characterization is spectral-free and applies in general Fréchet-space settings (Menet, 2013).

A later strengthening shows that operators satisfying the Frequent Hypercyclicity Criterion are hereditarily frequently hypercyclic. This hereditary frequent hypercyclicity is a reinforcement of frequent hypercyclicity tailored to direct-sum questions. In particular, the direct sum of a hereditarily frequently hypercyclic operator with any frequently hypercyclic operator is frequently hypercyclic. The same work proves that, on $X$8, a weighted backward shift is frequently hypercyclic if and only if it is hereditarily frequently hypercyclic, while also exhibiting frequently hypercyclic operators that are not hereditarily frequently hypercyclic, including a $X$9-type operator on $TSx=x$0 and a pair of frequently hypercyclic weighted shifts on $TSx=x$1 whose direct sum is not $TSx=x$2-frequently hypercyclic (Bayart et al., 2024).

An open problem remains whether there exists an operator on some Fréchet space that possesses a hypercyclic subspace but does not possess hereditarily hypercyclic subspaces, and in particular whether such an operator can be weakly mixing (Menet, 2013).

4. Invariant measures, strong mixing, and random models

The Frequent Hypercyclicity Criterion has a strong measure-theoretic content. If $TSx=x$3 acts on a separable $TSx=x$4-space and there exist a dense subset $TSx=x$5 and maps $TSx=x$6 such that for each $TSx=x$7,

$TSx=x$8

converge unconditionally, and

$TSx=x$9

then there exists a $T$0-invariant strongly mixing Borel probability measure on $T$1 with full support. For unilateral backward shifts on sequence $T$2-spaces, the construction can be sharpened to produce an exact invariant measure with full support (Murillo-Arcila et al., 2013).

A probabilistic realization of this phenomenon was later obtained through random series. If $T$3 for a dense bi-infinite family $T$4, and if $T$5 converges unconditionally for a suitable positive sequence $T$6, then a random vector

$T$7

with $T$8 i.i.d. and sufficiently light tails, is almost surely well defined and frequently hypercyclic for $T$9, and its distribution is a strongly mixing invariant measure with full support; in the unilateral case the measure is exact (Agneessens, 2022).

Applied to operators satisfying the Frequent Hypercyclicity Criterion, this yields an explicit random-series model. There exist a supercyclic vector $X$0, a sequence $X$1 with $X$2 and $X$3 for $X$4, and a random variable $X$5 with full support such that

$X$6

is almost surely frequently hypercyclic for $X$7 and induces a strongly mixing measure with full support. This recovers the Murillo–Peris existence theorem for strongly mixing invariant measures, but with an explicit random-vector model inside the space itself (Agneessens, 2022).

5. Quantitative interpretations and stronger density scales

The classical conclusion of FHC uses natural lower density, but later work showed that this understates the amount of recurrence forced by the criterion. Using admissible weighted densities $X$8 arising from non-negative regular summability matrices, operators satisfying FHC were shown to be $X$9-frequently hypercyclic and hence $x\in X$0-frequently hypercyclic for every $x\in X$1; by contrast, there is no $x\in X$2-frequently hypercyclic operator (Ernst et al., 2016).

A subsequent refinement sharpened the frequent universality conclusion even further. Under the Frequent Universality Criterion, one can obtain $x\in X$3-frequent universality and, more strongly, $x\in X$4-frequent universality for every $x\in X$5, as well as $x\in X$6-frequent universality for a broad class of slowly varying weights $x\in X$7. At the same time, no operator can be $x\in X$8-frequently hypercyclic, so these refinements are optimal within the density scale considered (Ernst et al., 2018).

These weighted-density results show that FHC defines a strict subclass of frequently hypercyclic operators. There exist frequently hypercyclic unilateral weighted shifts on $x\in X$9 that are not $S:X_0\to X_0$00-frequently hypercyclic for any $S:X_0\to X_0$01, and there also exists a logarithmically-frequently hypercyclic operator that is not frequently hypercyclic (Ernst et al., 2016, Ernst et al., 2018). Thus the criterion is not merely a convenient sufficient condition; it isolates a quantitatively stronger recurrence regime.

6. Variants, operator classes, and current extensions

Many classical operators are known to satisfy the Frequent Hypercyclicity Criterion, including certain weighted backward shifts on $S:X_0\to X_0$02 and $S:X_0\to X_0$03, translation operators on spaces of entire functions, and some composition operators and differential operators on suitable function spaces (Menet, 2013). Recent work has made this operator-level picture increasingly explicit.

For composition operators on the little Lipschitz space $S:X_0\to X_0$04, strictly increasing symbols $S:X_0\to X_0$05 with $S:X_0\to X_0$06 satisfy the Frequent Hypercyclicity Criterion exactly when the corresponding composition operator is hypercyclic, and this is equivalent to

$S:X_0\to X_0$07

The proof proceeds by conjugating the composition operator to an explicit block-sum operator on $S:X_0\to X_0$08 and then verifying the criterion there (López-Martínez, 5 May 2025).

For sequences of convolution operators $S:X_0\to X_0$09 on $S:X_0\to X_0$10, a symbol-level frequent hypercyclicity criterion has been obtained: if the $S:X_0\to X_0$11 are entire of exponential type, non-vanishing on a common annulus, and satisfy explicit modulus comparison and summability conditions on the boundary circles, then $S:X_0\to X_0$12 is frequently hypercyclic. This realizes the Bonilla–Grosse-Erdmann sequence criterion through Borel transforms and Pólya-type integral formulas (Bernal-González et al., 22 Feb 2026).

For weighted composition $S:X_0\to X_0$13-semigroups on $S:X_0\to X_0$14, the Mangino–Peris semigroup version of the Frequent Hypercyclicity Criterion becomes equivalent to chaos under natural hypotheses on the vector field $S:X_0\to X_0$15 and weight $S:X_0\to X_0$16; analogous equivalences hold on invariant Sobolev subspaces $S:X_0\to X_0$17 (Kalmes, 2014). On the abstract side, the criterion has been generalized to $S:X_0\to X_0$18-Frequent Hypercyclicity Criteria for hypercyclicity sets (Bès et al., 2014), to $S:X_0\to X_0$19-hypercyclicity criteria for Furstenberg families and scalar sets (Alves et al., 2024), to common frequent universality criteria for countable families of operators (Charpentier et al., 2020), to $S:X_0\to X_0$20-Frequent Hypercyclicity Criteria based on $S:X_0\to X_0$21-lower density (Gupta et al., 2014), and to $S:X_0\to X_0$22-Frequent Hypercyclicity Criteria in which natural density is replaced by a calibration function $S:X_0\to X_0$23 (Carroll et al., 9 Jun 2026).

Across these variants, the defining pattern remains the same: a dense set of “test vectors,” a forward–backward decomposition governed by maps $S:X_0\to X_0$24 or $S:X_0\to X_0$25, and unconditional convergence strong enough to make orbit pieces interact predictably. The enduring importance of the Frequent Hypercyclicity Criterion lies in this combination of analytic rigidity and dynamical reach: it is simultaneously a proof device, a structural principle, and a point of departure for increasingly refined notions of linear recurrence.