Spectral Gap Property Research Articles

Exponential rate of decay of correlations of equilibrium states associated with non-uniformly expanding circle maps

In the context of expanding maps of the circle with an indifferent fixed point, understanding the joint behavior of dynamics and pairs of moduli of continuity (ω,Ω) may be a useful element for the development of equilibrium theory. Here we identify a particular feature of modulus Ω (precisely limx→0+supdΩ(dx)/Ω(d)=0 ) as a sufficient condition for the system to exhibit exponential decay of correlations with respect to the unique equilibrium state associated with a potential having ω as modulus of continuity. This result is derived from obtaining the spectral gap property for the transfer operator acting on the space of observables with Ω as modulus of continuity, a property that, as is well known, also ensures the Central Limit Theorem. Examples of application of our results include the Manneville–Pomeau family.

The BPHZ Theorem for Regularity Structures via the Spectral Gap Inequality

We provide a relatively compact proof of the BPHZ theorem for regularity structures of decorated trees in the case where the driving noise satisfies a suitable spectral gap property, as in the Gaussian case. This is inspired by the recent work (Linares et al. in A diagram-free approach to the stochastic estimates in regularity structures, 2021. arXiv:2112.10739) in the multi-index setting, but our proof relies crucially on a novel version of the reconstruction theorem for a space of “pointed Besov modelled distributions”. As a consequence, the analytical core of the proof is quite short and self-contained, which should make it easier to adapt the proof to different contexts (such as the setting of discrete models).

Thermodynamical and spectral phase transition for local diffeomorphisms in the circle

It is known that all uniformly expanding dynamics have no phase transition with respect to Hölder continuous potentials. In this paper we show that given a local diffeomorphism f on the circle, that is neither a uniformly expanding dynamics nor invertible, the topological pressure function is not analytical. In other words, f has a thermodynamic phase transition with respect to geometric potential. Assuming that f is transitive and that Df is Hölder continuous, we show that there exists such that the transfer operator , acting on the space of Hölder continuous functions, has the spectral gap property for all and has not the spectral gap property for all . Similar results are also obtained when the transfer operator acts on the space of bounded variations functions and smooth functions. In particular, we show that in the transitive case f has a unique thermodynamic phase transition and it occurs in t 0. In addition, if the loss of expansion of the dynamics occurs because of an indifferent fixed point or the dynamics admits an absolutely continuous invariant probability with positive Lyapunov exponent then

Spectral gap and strict outerness for actions of locally compact groups on full factors

We prove that an outer action of a locally compact group G on a full factor M is automatically strictly outer, meaning that the relative commutant of M in the crossed product is trivial. If moreover the image of G in the outer automorphism group Out M is closed, we prove that the crossed product remains full. We obtain this result by proving that the inclusion of M in the crossed product automatically has a spectral gap property. Such results had only been proven for actions of discrete groups and for actions of compact groups, by using quite different methods in both cases. Even for the canonical free Bogoljubov actions on free group factors or free Araki–Woods factors, these results are new.

Aldous’ spectral gap property for normal Cayley graphs on symmetric groups

Aldous’ spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on Sn with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of Sn. In this paper, we prove three results on normal Cayley graphs on Sn possessing this property for sufficiently large n, one of which can be viewed as a generalization of the “normal” case of Aldous’ spectral gap conjecture.

On the Robin spectrum for the equilateral triangle* *Dedicated to Michael Berry for his 80th birthday.

The equilateral triangle is one of the few planar domains where the Dirichlet and Neumann eigenvalue problems were explicitly determined, by Lamé in 1833, despite not admitting separation of variables. In this paper, we study the Robin spectrum of the equilateral triangle, which was determined by McCartin in 2004 in terms of a system of transcendental coupled secular equations. We give uniform upper bounds for the Robin–Neumann gaps, showing that they are bounded by their limiting mean value, which is hence an almost sure bound. The spectrum admits a systematic double multiplicity, and after removing it we study the gaps in the resulting desymmetrized spectrum. We show a spectral gap property, that there are arbitrarily large gaps, and also arbitrarily small ones, moreover that the nearest neighbour spacing distribution of the desymmetrized spectrum is a delta function at the origin. We show that for sufficiently small Robin parameter, the desymmetrized spectrum is simple.

On the dimension of the space of harmonic functions on transitive shift spaces

In this paper, we show a new relation between phase transition in Statistical Mechanics and the dimension of the space of harmonic functions (SHF) for a transfer operator. This is accomplished by extending the classical Ruelle-Perron-Frobenius theory to the realm of low regular potentials defined on either finite or infinite (uncountable) alphabets. We also give an example of a potential having a phase transition where the Perron-Frobenius eigenvector space has dimension two. We discuss entropy and equilibrium states, in this general setting, and show that if the SHF is non-trivial, then the associated equilibrium states have full support. We also obtain a weak invariance principle in cases where the spectral gap property is absent. As a consequence, a functional central limit theorem for non-local observables of the Dyson model is obtained.

Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity

<p style='text-indent:20px;'>We consider a wide family of non-uniformly expanding maps and hyperbolic Hölder continuous potentials. We prove that the unique equilibrium state associated to each element of this family is given by the eigenfunction of the transfer operator and the eigenmeasure of the dual operator (both having the spectral radius as eigenvalue). We show that the transfer operator has the spectral gap property in some space of Hölder continuous observables and from this we obtain an exponential decay of correlations and a central limit theorem for the equilibrium state. Moreover, we establish the analyticity with respect to the potential of the equilibrium state as well as that of other thermodynamic quantities. Furthermore, we derive similar results for the equilibrium state associated to a family of non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials.

Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails

On a measure theoretical dynamical system with spectral gap property we consider non-integrable observables with regularly varying tails. Under a mild mixing condition we show that the appropriately normed and trimmed sum process of these observables then converges in mean. This result is new also for the special case of i.i.d. random variables and contrasts the general case where mean convergence might fail even though a strong law of large numbers holds. To illuminate the required mixing condition we give an explicit example of a dynamical system fulfilling a spectral gap property and an observable with regularly varying tails but without the assumed mixing condition such that mean convergence fails.

Spectral Gap Property for Random Dynamics on the Real Line and Multifractal Analysis of Generalised Takagi Functions

We consider the random iteration of finitely many expanding $\mathcal{C}^{1+\epsilon}$ diffeomorphisms on the real line without a common fixed point. We derive the spectral gap property of the associated transition operator acting on H\"older spaces. As an application we introduce generalised Takagi functions on the real line and we perform a complete multifractal analysis of the pointwise H\"older exponents of these functions.

Spectral theory and time asymptotics of size-structured two-phase population models

This work provides a general spectral analysis of size-structured two-phase population models. Systematic functional analytic results are given. We deal first with the case of finite maximal size. We characterize the irreducibility of the corresponding $ L^{1} $ semigroup in terms of properties of the different parameters of the system. We characterize also the spectral gap property of the semigroup. It turns out that the irreducibility of the semigroup implies the existence of the spectral gap. In particular, we provide a general criterion for asynchronous exponential growth. We show also how to deal with time asymptotics in case of lack of irreducibility. Finally, we extend the theory to the case of infinite maximal size.

Quasi-regular representations of discrete groups and associated $C^*$-algebras

Let G be a countable group. We introduce several equivalence relations on the set Sub(G) of subgroups of G, defined by properties of the quasi-regular representations λ G/H associated to H ∈ Sub(G) and compare them to the relation of G-conjugacy of subgroups. We define a class Sub sg (G) of subgroups (these are subgroups with a certain spectral gap property) and show that they are rigid, in the sense that the equivalence class of H ∈ Sub sg (G) for any one of the above equivalence relations coincides with the G-conjugacy class of H. Next, we introduce a second class Sub w−par (G) of subgroups (these are subgroups which are weakly parabolic in some sense) and we establish results concerning the ideal structure of the C

Local-global principles in circle packings

We generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral Apollonian gaskets, Invent. Math.196(2014), 589–650] and Zhang [On the local-global principle for integral Apollonian 3-circle packings, J. Reine Angew. Math.737, (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$satisfying certain conditions, where$K$is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that${\mathcal{A}}$possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$containing a Zariski dense subgroup of$\operatorname{PSL}_{2}(\mathbb{Z})$.

Asymptotic behaviour of neuron population models structured by elapsed-time

We study two population models describing the dynamics of interacting neurons, initially proposed by Pakdaman et al (2010 Nonlinearity 23 55–75) and Pakdaman et al (2014 J. Math. Neurosci. 4 1–26). In the first model, the structuring variable s represents the time elapsed since its last discharge, while in the second one neurons exhibit a fatigue property and the structuring variable is a generic ‘state’. We prove existence of solutions and steady states in the space of finite, nonnegative measures. Furthermore, we show that solutions converge to the equilibrium exponentially in time in the case of weak nonlinearity (i.e. weak connectivity). The main innovation is the use of Doeblin’s theorem from probability in order to show the existence of a spectral gap property in the linear (no-connectivity) setting. Relaxation to the steady state for the nonlinear models is then proved by a constructive perturbation argument.

Strong laws of large numbers for intermediately trimmed Birkhoff sums of observables with infinite mean

We consider dynamical systems on a finite measure space fulfilling a spectral gap property and Birkhoff sums of a non-negative, non-integrable observable. For such systems we generalize strong laws of large numbers for intermediately trimmed sums only known for independent random variables. The results split up in trimming statements for general distribution functions and for regularly varying tail distributions. In both cases the trimming rate can be chosen in the same or almost the same way as in the i.i.d. case. As an example we show that piecewise expanding interval maps fulfill the necessary conditions for our limit laws. As a side result we obtain strong laws of large numbers for truncated Birkhoff sums.

Spectral gap property and strong ergodicity for groups of affine transformations of solenoids

Let $X$ be a solenoid, i.e. a compact, finite-dimensional, connected abelian group with normalized Haar measure $\unicode[STIX]{x1D707}$, and let $\unicode[STIX]{x1D6E4}\rightarrow \operatorname{Aff}(X)$ be an action of a countable discrete group $\unicode[STIX]{x1D6E4}$ by continuous affine transformations of $X$. We show that the probability measure preserving action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ does not have the spectral gap property if and only if there exists a $p_{\text{a}}(\unicode[STIX]{x1D6E4})$-invariant proper subsolenoid $Y$ of $X$ such that the image of $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually solvable group, where $p_{\text{a}}:\operatorname{Aff}(X)\rightarrow \operatorname{Aut}(X)$ is the canonical projection. When $\unicode[STIX]{x1D6E4}$ is finitely generated or when $X$ is the $a$-adic solenoid for an integer $a\geq 1$, the subsolenoid $Y$ can be chosen so that the image $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually abelian group. In particular, an action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ by affine transformations on a solenoid $X$ has the spectral gap property if and only if $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is strongly ergodic.

Passive tracer in non-Markovian, Gaussian velocity field

We consider the trajectory of a tracer that is the solution of an ordinary differential equation Ẋ(t)=V(t,X(t)),X(0)=0, with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible realizations. It is known, see Fannjiang and Komorowski (1999), Carmona and Xu (1996) and Komorowski et al. (2012), that X(t)∕t converges in law, as t→+∞, to a normal, zero mean vector, provided that the field V(t,x) is Markovian and has the spectral gap property. We wish to extend this result to the case when the field is not Markovian and its covariance matrix is given by a completely monotone Bernstein function.

Superexpanders from group actions on compact manifolds

It is known that the expanders arising as increasing sequences of level sets of warped cones, as introduced by the second-named author, do not coarsely embed into a Banach space as soon as the corresponding warped cone does not coarsely embed into this Banach space. Combining this with non-embeddability results for warped cones by Nowak and Sawicki, which relate the non-embeddability of a warped cone to a spectral gap property of the underlying action, we provide new examples of expanders that do not coarsely embed into any Banach space with nontrivial type. Moreover, we prove that these expanders are not coarsely equivalent to a Lafforgue expander. In particular, we provide infinitely many coarsely distinct superexpanders that are not Lafforgue expanders. In addition, we prove a quasi-isometric rigidity result for warped cones.

Fullness and Connes' τ invariant of type III tensor product factors

We show that the tensor product M⊗‾N of any two full factors M and N (possibly of type III) is full and we compute Connes' invariant τ(M⊗‾N) in terms of τ(M) and τ(N). The key novelty is an enhanced spectral gap property for full factors of type III. Moreover, for full factors of type III with almost periodic states, we prove an optimal spectral gap property. As an application of our main result, we also show that for any full factor M and any non-type I amenable factor P, the tensor product factor M⊗‾P has a unique McDuff decomposition, up to stable unitary conjugacy.

Strongly ergodic actions have local spectral gap

We show that an ergodic measure preserving action Γ ↷ ( X , μ ) \Gamma \curvearrowright (X,\mu ) of a discrete group Γ \Gamma on a σ \sigma -finite measure space ( X , μ ) (X,\mu ) satisfies the local spectral gap property introduced in Invent. Math. 208 (2017), 715–802, if and only if it is strongly ergodic. In fact, we prove a more general local spectral gap criterion in arbitrary von Neumann algebras. Using this criterion, we also obtain a short proof of Connes’ spectral gap theorem for full I I 1 \mathrm {II}_1 factors as well as its recent generalization to full type I I I \mathrm {III} factors.