$ρ$-Frequently Hypercyclic Operators

Abstract: The concept of $ρ$-frequent hypercyclicity is introduced in order to provide a refined form of frequent hypercyclicity. This is achieved by replacing the denominator in the definition of frequent hypercyclicity by an appropriately chosen calibration function $ρ$. A $ρ$-Frequent Hypercyclicity Criterion is determined and the $ρ$-frequent hypercyclicity of weighted backward shifts is investigated.

• The paper introduces ho-frequency hypercyclicity, defined by a characterization asserting positive lower ho-density with a limit infimum, adding structure-based method for measuring how frequently a bounded backward shift hypercycles, agreeing with known ergodic continuous function properties and density calculus.
• Researchers demonstrated a $ ho$-Frequent Hypercyclicity Criterion, extending the Frequent Universality Criterion to $ ho$-frequencies of hypercyclic operators, applicable for separating slopes in logarithmic especial specifics as Bayart-Grivaux and Ernst-Mouze.
• The theory applies to weighted backward shifts on $\ell^p(\mathbb{N})$, showing a shift to be $\rho$-frequently hypercyclic under given uniform conditions for weights; an open question concerns necessity of the uniform condition, potentially connecting back to comparing Erdős–Sárközy and Vandiver dense/frequent return sets.

The paper introduces $\rho$-frequent hypercyclicity, a refinement of frequent hypercyclicity in linear dynamics obtained by replacing the natural density in Bayart and Grivaux's original definition with a lower density calibrated by a slowly growing function $\rho$. The framework interpolates between ordinary hypercyclicity and frequent hypercyclicity, yields a $\rho$-Frequent Hypercyclicity Criterion, produces a necessary and sufficient condition for weighted backward shifts on $\ell^p(\mathbb{N})$ under a weight restriction, and demonstrates that distinct calibration functions give genuinely distinct notions of hypercyclicity.

Densities relative to a calibration function

A calibration function $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$ is continuous, strictly increasing, unbounded, Lip$(1,1)$, concave, and has non-increasing $\rho(t)/t$; these properties need only hold for large $t$, and examples include $t^\alpha$ ($0<\alpha<1$), $\rho$0, $\rho$1, and $\rho$2. The lower $\rho$3-density of a set $\rho$4 is

$\rho$5

which recovers the classical lower density when $\rho$6. A basic lemma characterizes positive lower $\rho$7-density for a strictly increasing sequence $\rho$8: it holds if and only if $\rho$9 as $\rho$0, extending the familiar condition $\rho$1.

Central to the development is the sequence $\rho$2, which the authors identify as the canonical sequence of $\rho$3-density 1. Two structural facts about $\rho$4 carry the later arguments: it is strictly increasing, and it satisfies the convexity-type inequality $\rho$5, a consequence of concavity and the monotonicity of $\rho$6.

The $\rho$7-Frequent Hypercyclicity Criterion

An operator $\rho$8 on a separable infinite-dimensional Fréchet space is $\rho$9-frequently hypercyclic if some vector's return set to every nonempty open set has positive lower $\ell^p(\mathbb{N})$0-density. The main criterion generalizes the Frequent Universality Criterion of Bonilla and Grosse-Erdmann: given a dense set $\ell^p(\mathbb{N})$1 and maps $\ell^p(\mathbb{N})$2 satisfying unconditional convergence conditions along the subsequence $\ell^p(\mathbb{N})$3 — namely uniform unconditional convergence of $\ell^p(\mathbb{N})$4 and $\ell^p(\mathbb{N})$5, unconditional convergence of $\ell^p(\mathbb{N})$6, and $\ell^p(\mathbb{N})$7 — the sequence $\ell^p(\mathbb{N})$8 is $\ell^p(\mathbb{N})$9-frequently universal.

The proof is a reduction rather than an independent construction: the rescaled sequences satisfy the hypotheses of the original Frequent Universality Criterion, producing return times $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$0, and since $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$1, the return set contains a sequence of positive lower $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$2-density. Specializing to iterates of a single operator gives the $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$3-Frequent Hypercyclicity Criterion with conditions expressed through powers $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$4 and $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$5.

Weighted backward shifts on $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$6

For a weighted backward shift $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$7 on $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$8, $\rho\colon\mathbb{R}^+\to\mathbb{R}^+$9, with weights $(1,1)$0 and $(1,1)$1, the criterion yields a clean sufficient condition: $(1,1)$2 is $(1,1)$3-frequently hypercyclic whenever, for each $(1,1)$4,

$(1,1)$5

converges uniformly in $(1,1)$6. Conditions ($(1,1)$7-i) and ($(1,1)$8-iii) hold automatically for shifts, so only this series condition matters. In the case $(1,1)$9 it reduces to $\rho(t)/t$0, known to be sufficient by Bayart–Grivaux and necessary by Bayart–Ruzsa via their extension of the Erdős–Sárközy difference-set theorem.

The paper establishes necessity of a matching condition for shifts with all weights $\rho(t)/t$1: $\rho(t)/t$2 is $\rho(t)/t$3-frequently hypercyclic if and only if

$\rho(t)/t$4

Necessity combines two ingredients: a standard proposition showing that any $\rho(t)/t$5-frequently hypercyclic shift admits a positive-lower-$\rho(t)/t$6-density sequence $\rho(t)/t$7 with $\rho(t)/t$8, and a Cauchy condensation argument showing that convergence along such a sequence forces convergence along the canonical sequence $\rho(t)/t$9. This result is the engine behind the separation examples below. The authors are explicit that the necessity of the full uniform condition for general weighted shifts remains open; the equivalence here relies on the restriction $t$0, which makes $t$1 non-increasing.

Separation across calibration functions

The theory would be vacuous if all calibration functions induced the same notion. Theorem on varying $t$2 shows they do not: if $t$3 where $t$4 is increasing and unbounded, there is a bounded weighted backward shift on $t$5 that is $t$6-frequently hypercyclic but not $t$7-frequently hypercyclic. For instance, there exists a bounded weighted backward shift that is $t$8-frequently hypercyclic but not frequently hypercyclic.

The construction sets most weights to 1 and assigns $t$9 at indices $t^\alpha$0, where $t^\alpha$1 comes from a combinatorial lemma guaranteeing $t^\alpha$2, at worst geometric decay $t^\alpha$3, divergence of $t^\alpha$4, and convergence of $t^\alpha$5. A counting estimate shows that between consecutive elements of $t^\alpha$6 the faster function $t^\alpha$7 crosses at least roughly $t^\alpha$8 integers of the form $t^\alpha$9, so the divergence of $0<\alpha<1$0 forces $0<\alpha<1$1 while $0<\alpha<1$2. These examples are analogues of results of Ernst–Mouze, who produced operators that are logarithmically frequently hypercyclic but not frequently hypercyclic within their admissible-matrix framework.

Relation to existing formulations

The paper carefully situates $0<\alpha<1$3-frequent hypercyclicity against the $0<\alpha<1$4-frequent hypercyclicity of Bayart–Matheron, where return sets must be enumerable as sequences with $0<\alpha<1$5, and against $0<\alpha<1$6-frequent hypercyclicity of Gupta–Mundayadan (the case $0<\alpha<1$7). The two viewpoints are dual: $0<\alpha<1$8-density counts how often the orbit visits $0<\alpha<1$9 up to time $\rho$00, while $\rho$01-boundedness measures how long one waits for the $\rho$02-th visit. They are not equivalent in general — the obstruction being slow growth of $\rho$03 or fast growth of $\rho$04 — and the paper gives explicit counterexamples in both directions using $\rho$05 and $\rho$06.

Equivalence does hold under doubling-type regularity. If $\rho$07 eventually, then positive lower $\rho$08-density coincides with $\rho$09-boundedness for $\rho$10; conversely, any increasing $\rho$11 with non-decreasing gaps and $\rho$12 generates, by piecewise-linear interpolation, a calibration function whose $\rho$13-density captures exactly the $\rho$14-bounded sequences. Consequently $\rho$15-frequent hypercyclicity is precisely $\rho$16-frequent hypercyclicity for $\rho$17, and the Gupta–Mundayadan separation theorem for $\rho$18 becomes a special case of the general separation theorem. An illustrative example gives weights $\rho$19, yielding a shift on $\rho$20 that is $\rho$21-frequently hypercyclic but not $\rho$22-frequently hypercyclic for any $\rho$23.

Finally, the collection of sets of positive lower $\rho$24-density forms a proper Furstenberg family, so $\rho$25-frequent hypercyclicity is a special case of $\rho$26-hypercyclicity (Bès–Menet–Peris–Puig) and of Kostić's $\rho$27-hypercyclicity.

Limitations and open questions

Two limitations are acknowledged explicitly. First, the necessary and sufficient condition $\rho$28 is proved only for shifts with $\rho$29; whether the uniform sufficient condition is necessary for arbitrary weighted backward shifts on $\rho$30 is left open, and its resolution for $\rho$31 required the substantial Erdős–Sárközy-type machinery of Bayart–Ruzsa, suggesting the general case may be difficult. Second, the separation examples are constructed on $\rho$32 rather than general $\rho$33, though the authors note straightforward modifications should extend them. The relationship between $\rho$34-density and $\rho$35-boundedness also remains incomplete outside the doubling regimes covered by the equivalence proposition.

Conclusion

The paper provides a coherent calibration-function framework for measuring how frequently a hypercyclic orbit returns to open sets, complete with a criterion adapted from Bonilla–Grosse-Erdmann, a sharp characterization for a natural class of weighted backward shifts, and concrete evidence that the resulting hierarchy of hypercyclicity notions is strict. Its main open problem — necessity of the uniform series condition for unrestricted weights — is the natural next target for this line of work.