Infinite measure renewal theorem and related results

Perturbation approach to scaled type Markov renewal processes with infinite mean

Scaled type Markov renewal processes generalize classical renewal processes: renewal times come from a one parameter family of probability laws and the sequence of the parameters is the trajectory of an ergodic Markov chain. Our primary interest here is the asymptotic distribution of the Markovian parameter at time t \to \infty. The limit, of course, depends on the stationary distribution of the Markov chain. The results, however, are essentially different depending on whether the expectations of the renewals are finite or infinite. If the expectations are uniformly bounded, then we can provid…

Operator renewal theory and mixing rates for dynamical systems with infinite measure

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\…

Fourier decay, Renewal theorem and Spectral gaps for random walks on split semisimple Lie groups

We establish an exponential error term for the renewal theorem in the context of products of random matrices, which is surprising compared with classical abelian cases. A key tool is the Fourier decay of the Furstenberg measures on the projective spaces, which is a higher dimensional generalization of a recent work of Bourgain-Dyatlov.

Darling-Kac theorem for renewal shifts in the absence of regular variation

We study null recurrent renewal Markov chains with renewal distribution in the domain of geometric partial attraction of a semistable law. Using the classical procedure of inversion, we derive a limit theorem similar to the Darling-Kac law along subsequences and obtain some interesting properties of the limit distribution. Also in this context, we obtain a Karamata type theorem along subsequences for positive operators. In both results, we identify the allowed class of subsequences. We provide several examples of nontrivial infinite measure preserving systems to which these results apply.

The rate of convergence for the renewal theorem in $\mathbb{R}^d$

Let $ρ$ be a borelian probability measure on $\mathrm{SL}_d(\mathbb{R})$. Consider the random walk $(X_n)$ on $\mathbb{R}^d\setminus\{0\}$ defined by $ρ$ : for any $x\in \mathbb{R}^d\setminus\{0\}$, we set $X_0 =x$ and $X_{n+1} = g_{n+1} X_n$ where $(g_n)$ is an iid sequence of $\mathrm{SL}_d(\mathbb{R})-$valued random variables of law $ρ$. Guivarc'h and Raugi proved that under an assumption on the subgroup generated by the support of $ρ$ (strong irreducibility and proximality), this walk is transient. In particular, this proves that if $f$ is a compactly supported continuous function on $\mat…

Strong renewal theorems and Lyapunov spectra for $α$-Farey and $α$-Lüroth systems

In this paper we introduce and study the $α$-Farey map and its associated jump transformation, the $α$-Lüroth map, for an arbitrary countable partition $α$ of the unit interval with atoms which accumulate only at the origin. These maps represent linearised generalisations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic-theoretic properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called $α$…

S-间隙移位上更新型势函数的有效内在遍历性

Effective Intrinsic Ergodicity for renewal-type potentials on S-gap shifts

针对S-间隙移位上的更新型势函数，通过诱导到全移位并利用更新方程，证明了唯一平衡态的存在性，并给出了有效内在遍历性估计：自由能在压力Δ内的不变测度与平衡态在Hölder观测下的距离为O(√Δ)。

On mixing and the local central limit theorem for hyperbolic flows

We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal processes, Axiom A flows, as well as the systems admitting Young tower, such as Sinai billiard with finite horizon, suspensions over Pomeau-Manneville maps, and geometric Lorenz attractors.

Mixing for invertible infinite measure systems

In a recent paper, Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math. 189 (2012), 61-110] obtained results on mixing and mixing rates for a large class of noninvertible maps preserving an infinite ergodic invariant measure. Here, we are concerned with extending these results to the invertible setting. Mixing is established for a large class of infinite measure invertible maps. Assuming additional structure, in particular exponential contraction along stable manifolds, it is possible to obtain good results on mixing rates …

Mixing for continuous time dynamical systems with infinite measure

We develop operator renewal theory for flows and apply this to infinite ergodic theory. In particular we obtain results on mixing for a large class of infinite measure semiflows. Examples of systems covered by our results include suspensions over parabolic rational maps of the complex plane, and nonuniformly expanding semiflows with indifferent periodic orbits. In some cases, we obtain higher order asymptotics and rates of mixing.

Thermodynamic Formalism for Generalized Countable Markov Shifts

Countable Markov shifts, denoted by $Σ_A$ for a 0-1 infinite matrix $A$, are central objects in symbolic dynamics and ergodic theory. R. Exel and M. Laca introduced the corresponding operator algebras, a generalization of the Cuntz-Krieger algebras for infinite countable alphabet, and the set $X_A$, a kind of Generalized Markov Shift (GMS) that coincides with $Σ_A$ in the locally compact case. The set $Σ_A$ is dense in $X_A$, and its complement, a set of finite allowed words, is dense in $X_A$ when non-empty. We develop the thermodynamic formalism for $X_A$, introducing the notion of conformal…

无穷遍历变换的约束停留时间

Tied-down occupation times of infinite ergodic transformations

该研究针对一类弱混合逐点对偶遍历变换，证明其游程后约束时刻停留时间的分布极限定理，提出约束更新混合性质以细化有理弱混合，并建立Gibbs-Markov与AFU映射的周期局部极限定理。

无限遍历变换束缚停留时间的广义均匀定律

Generalized uniform laws for tied-down occupation times of infinite ergodic transformations

本文利用算子更新理论建立无限遍历变换在轨道终点返回有限测度集条件下停留时间的条件极限定理，所得广义均匀极限定律适用于含多个中性不动点的间歇性映射。

Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts

We consider the natural definition of DLR measure in the setting of $σ$-finite measures on countable Markov shifts. We prove that the set of DLR measures contains the set of conformal measures associated with Walters potentials. In the BIP case, or when the potential normalizes the Ruelle's operator, we prove that the notions of DLR and conformal coincide. On the standard renewal shift, we study the problem of describing the cases when the set of the eigenmeasures jumps from finite to infinite measures when we consider high and low temperatures, respectively. For this particular shift, we prov…