Infinite measure preserving flows with infinite ergodic index

Alexandre I. Danilenko; Anton V. Solomko

Displaying similar documents to “Infinite measure preserving flows with infinite ergodic index”

Dispersing cocycles and mixing flows under functions

Klaus Schmidt (2002)

Fundamenta Mathematicae

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Let T be a measure-preserving and mixing action of a countable abelian group G on a probability space (X,,μ) and A a locally compact second countable abelian group. A cocycle c: G × X → A for T disperses if $l i m_{g \to \infty} c (g, \cdot) - α (g) = \infty$ in measure for every map α: G → A. We prove that such a cocycle c does not disperse if and only if there exists a compact subgroup A₀ ⊂ A such that the composition θ ∘ c: G × X → A/A₀ of c with the quotient map θ: A → A/A₀ is trivial (i.e. cohomologous to a homomorphism η: G → A/A₀). This...

Mixing via families for measure preserving transformations

Rui Kuang, Xiangdong Ye (2008)

Colloquium Mathematicae

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In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any...

On embeddability of automorphisms into measurable flows from the point of view of self-joining properties

Joanna Kułaga-Przymus (2015)

Fundamenta Mathematicae

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We compare self-joining and embeddability properties. In particular, we prove that a measure preserving flow ${(T_{t})}_{t \in ℝ}$ with T₁ ergodic is 2-fold quasi-simple (resp. 2-fold distally simple) if and only if T₁ is 2-fold quasi-simple (resp. 2-fold distally simple). We also show that the Furstenberg-Zimmer decomposition for a flow ${(T_{t})}_{t \in ℝ}$ with T₁ ergodic with respect to any flow factor is the same for ${(T_{t})}_{t \in ℝ}$ and for T₁. We give an example of a 2-fold quasi-simple flow disjoint from simple flows and whose time-one...

Example of a mean ergodic L¹ operator with the linear rate of growth

Wojciech Kosek (2011)

Colloquium Mathematicae

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The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $l i m_{n \to \infty} | | T ⁿ | | / n^{1 - γ} = \infty$ . In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $l i m s u p | | T ⁿ | | / n^{1 - γ ₀} = 0 .$ In this note we construct an example of a nonpositive L¹ operator with the...

Hamiltonian loops from the ergodic point of view

Leonid Polterovich (1999)

Journal of the European Mathematical Society

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Let $G$ be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $Y$ . A loop $h : S^{1} \to G$ is called strictly ergodic if for some irrational number the associated skew product map $T : S^{1} \times Y \to S^{1} \times Y$ defined by $T (t, y) = (t + α; h (t) y)$ is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of $G$ can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply...

Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory

Christophe Cuny (2010)

Studia Mathematica

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Let X be a closed subspace of $L^{p} (μ)$ , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $s u p_{n \in ℤ} | | U ⁿ | | < \infty$ . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like $\sum_{n \geq 1} (U ⁿ f) / n^{1 - α}$ , 0 ≤ α < 1, in terms of $| | f + \dots + U^{n - 1} f {| |}_{p}$ , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. ...

Non-Typical Points for β-Shifts

David Färm, Tomas Persson (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study sets of non-typical points under the map $f_{β} \mapsto β x$ mod 1 for non-integer β and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition β > 1.541 found in the above paper can be removed.

Positive L¹ operators associated with nonsingular mappings and an example of E. Hille

Isaac Kornfeld, Wojciech Kosek (2003)

Colloquium Mathematicae

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E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $| | T ⁿ | | \sim n^{1 / 4}$ ). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $l i m_{n \to \infty} | | T ⁿ | | / n^{1 - γ} = \infty . I n t h e c l a s s o f p o s i t i v e o p e r a t o r s t h e s e e x a m p l e s a r e t h e b e s t p o s s i b l e i n t h e s e n s e t h a t f o r e v e r y s u c h o p e r a t o r T t h e r e e x i s t s a γ ₀ > 0 s u c h t h a t$ lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0 $.$ A class of numerical sequences αₙ, intimately...

Polynomial decay of correlations for a class of smooth flows on the two torus

Bassam Fayad (2001)

Bulletin de la Société Mathématique de France

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Kočergin introduced in 1975 a class of smooth flows on the two torus that are mixing. When these flows have one fixed point, they can be viewed as special flows over an irrational rotation of the circle, with a ceiling function having a power-like singularity. Under a Diophantine condition on the rotation’s angle, we prove that the special flows actually have a $t^{- η}$ -speed of mixing, for some $η > 0$ .

On the ergodic decomposition for a cocycle

Jean-Pierre Conze, Albert Raugi (2009)

Colloquium Mathematicae

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Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure $m_{G}$ . We consider the map $τ_{φ}$ defined on X × G by $τ_{φ} : (x, g) \mapsto (τ x, φ (x) g)$ and the cocycle ${(φ ₙ)}_{n \in ℤ}$ generated by φ. Using a characterization of the ergodic invariant measures for $τ_{φ}$ , we give the form of the ergodic decomposition of $μ (d x) \otimes m_{G} (d g)$ or more generally of the $τ_{φ}$ -invariant measures $μ_{χ} (d x) \otimes χ (g) m_{G} (d g)$ , where $μ_{χ} (d x)$ is χ∘φ-conformal for an exponential χ on G.

Multiple disjointness and invariant measures on minimal distal flows

Juho Rautio (2015)

Studia Mathematica

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We examine multiple disjointness of minimal flows, that is, we find conditions under which the product of a collection of minimal flows is itself minimal. Our main theorem states that, for a collection ${X_{i}}_{i \in I}$ of minimal flows with a common phase group, assuming each flow satisfies certain structural and algebraic conditions, the product $\prod_{i \in I} X_{i}$ is minimal if and only if $\prod_{i \in I} X_{i}^{e q}$ is minimal, where $X_{i}^{e q}$ is the maximal equicontinuous factor of $X_{i}$ . Most importantly, this result holds when each $X_{i}$ is distal. When...

On invariant measures for piecewise $C^{2}$ -transformations of the n-dimensional cube

M. Jabłoński (1983)

Annales Polonici Mathematici

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Generalizations of Cesàro means and poles of the resolvent

Laura Burlando (2004)

Studia Mathematica

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An improvement of the generalization-obtained in a previous article [Bu1] by the author-of the uniform ergodic theorem to poles of arbitrary order is derived. In order to answer two natural questions suggested by this result, two examples are also given. Namely, two bounded linear operators T and A are constructed such that $n^{- 2} T ⁿ$ converges uniformly to zero, the sum of the range and the kernel of 1-T being closed, and $n^{- 3} \sum_{k = 0}^{n - 1} A^{k}$ converges uniformly, the sum of the range of 1-A and the kernel of (1-A)²...

The one-sided ergodic Hilbert transform in Banach spaces

Guy Cohen, Christophe Cuny, Michael Lin (2010)

Studia Mathematica

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Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hilbert transform $l i m_{n} \sum_{k = 1}^{n} (T^{k} x) / k$ . We prove that weak and strong convergence are equivalent, and in a reflexive space also $s u p_{n} | | \sum_{k = 1}^{n} (T^{k} x) / k | | < \infty$ is equivalent to the convergence. We also show that $- \sum_{k = 1}^{\infty} (T^{k}) / k$ (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup ${(I - T)}_{| \bar{(I - T) X}}^{r}$ .

Bohr local properties of $C_{Λ} (T)$

Françoise Lust-Piquard (1989)

Colloquium Mathematicae

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Almost everywhere convergence of generalized ergodic transforms for invertible power-bounded operators in $L^{p}$

Christophe Cuny (2011)

Colloquium Mathematicae

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We show that some results of Gaposhkin about a.e. convergence of series associated to a unitary operator U acting on L²(X,Σ,μ) (μ is a σ-finite measure) may be extended to the case where U is an invertible power-bounded operator acting on $L^{p} (X, Σ, μ)$ , p > 1. The proofs make use of the spectral integration initiated by Berkson-Gillespie and, more specifically, of recent results of the author.

Measure-preserving homeomorphisms of noncompact manifolds and mass flow toward ends

Tatsuhiko Yagasaki (2007)

Fundamenta Mathematicae

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Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and $ℋ_{E} (M, ω)$ the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism $c^{ω} : ℋ_{E} (M, ω) \to (M, ω)$ , which measures for each $h \in ℋ_{E} (M, ω)$ the mass flow toward ends induced by h. We show that the...

On the (C,α) Cesàro bounded operators

Elmouloudi Ed-dari (2004)

Studia Mathematica

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For a given linear operator T in a complex Banach space X and α ∈ ℂ with ℜ (α) > 0, we define the nth Cesàro mean of order α of the powers of T by $M ₙ^{α} = {(A ₙ^{α})}^{- 1} \sum_{k = 0}^{n} A_{n - k}^{α - 1} T^{k}$ . For α = 1, we find $M ₙ ¹ = {(n + 1)}^{- 1} \sum_{k = 0}^{n} T^{k}$ , the usual Cesàro mean. We give necessary and sufficient conditions for a (C,α) bounded operator to be (C,α) strongly (weakly) ergodic.

Pointwise convergence for subsequences of weighted averages

Patrick LaVictoire (2011)

Colloquium Mathematicae

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We prove that if μₙ are probability measures on ℤ such that μ̂ₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence $n_{k}$ such that the weighted ergodic averages corresponding to $μ_{n_{k}}$ satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ for a slowly growing function ρ. Under some monotonicity assumptions,...

On the convergence to 0 of mₙξmod 1

Bassam Fayad, Jean-Paul Thouvenot (2014)

Acta Arithmetica

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We show that for any irrational number α and a sequence ${m_{l}}_{l \in ℕ}$ of integers such that $l i m_{l \to \infty} | | | m_{l} α | | | = 0$ , there exists a continuous measure μ on the circle such that $l i m_{l \to \infty} \int_{} | | | m_{l} θ | | | d μ (θ) = 0$ . This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence ${m_{l}}_{l \in ℕ}$ of integers such that $| | | m_{l} α | | | \to 0$ and such that $m_{l} θ [1]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.

Ergodic theorems in fully symmetric spaces of τ-measurable operators

Vladimir Chilin, Semyon Litvinov (2015)

Studia Mathematica

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Junge and Xu (2007), employing the technique of noncommutative interpolation, established a maximal ergodic theorem in noncommutative $L_{p}$ -spaces, 1 < p < ∞, and derived corresponding maximal ergodic inequalities and individual ergodic theorems. In this article, we derive maximal ergodic inequalities in noncommutative $L_{p}$ -spaces directly from the results of Yeadon (1977) and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these...

An attraction result and an index theorem for continuous flows on $ℝ^{n} \times [0, \infty)$

Klaudiusz Wójcik (1997)

Annales Polonici Mathematici

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We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E = ℝ^{n + 1}$ for which $\partial E = ℝ^{n} \times 0$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on $ℝ^{n} \times [0, \infty)$ .

Large sets of integers and hierarchy of mixing properties of measure preserving systems

Vitaly Bergelson, Tomasz Downarowicz (2008)

Colloquium Mathematicae

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We consider a hierarchy of notions of largeness for subsets of ℤ (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βℤ to establish connections between various notions of largeness and apply those results to the study of the sets $R_{A, B}^{ε} = n \in ℤ : μ (A \cap T ⁿ B) > μ (A) μ (B) - ε$ of times of “fat intersection”. Among other things we show that the sets $R_{A, B}^{ε}$ allow one...

Distortion bounds for $C^{2 + η}$ unimodal maps

Mike Todd (2007)

Fundamenta Mathematicae

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We obtain estimates for derivative and cross-ratio distortion for $C^{2 + η}$ (any η > 0) unimodal maps with non-flat critical points. We do not require any “Schwarzian-like” condition. For two intervals J ⊂ T, the cross-ratio is defined as the value B(T,J): = (|T| |J|)/(|L| |R|) where L,R are the left and right connected components of T∖J respectively. For an interval map g such that $g_{T} : T \to ℝ$ is a diffeomorphism, we consider the cross-ratio distortion to be B(g,T,J): = B(g(T),g(J))/B(T,J). We prove...

Pointwise convergence of nonconventional averages

I. Assani (2005)

Colloquium Mathematicae

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We answer a question of H. Furstenberg on the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} U ⁿ (f \cdot R ⁿ (g))$ , where U and R are positive operators. We also study the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} f (S ⁿ x) g (R ⁿ x)$ when T and S are measure preserving transformations.

Topological dynamics of unordered Ramsey structures

Moritz Müller, András Pongrácz (2015)

Fundamenta Mathematicae

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We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M (G_{)}$ of the automorphism group $G$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of $M (G_{)}$ . We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M (G_{)}$ admits...

The max norm in $R^{n}$ -isometries and measure

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

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Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

Earl Berkson (2014)

Studia Mathematica

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Let $f \in V_{r} () \cup_{r} ()$ , where, for 1 ≤ r < ∞, $V_{r} ()$ (resp., $_{r} ()$ ) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values,...

Transference of weak type bounds of multiparameter ergodic and geometric maximal operators

Paul Hagelstein, Alexander Stokolos (2012)

Fundamenta Mathematicae

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Let $U ₁, . . ., U_{d}$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $ℤ ₊^{d}$ and the associated collection of rectangular parallelepipeds in $ℝ^{d}$ with sides parallel to the axes and dimensions of the form $n ₁ \times \dots \times n_{d}$ with $(n ₁, . . ., n_{d}) \in Γ .$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{Γ}$ are defined respectively on $L ¹ (ℝ^{d})$ and L¹(Ω) by $M_{} g (x) = s u p_{x \in R \in} 1 / | R | \int_{R} | g (y) | d y$ and $M_{Γ} f (ω) = s u p_{(n ₁, . . ., n_{d}) \in Γ} 1 / n ₁ \dots n_{d} \sum_{j ₁ = 0}^{n ₁ - 1} \dots \sum_{j_{d} = 0}^{n_{d} - 1} | f (U ₁^{j ₁} \dots U_{d}^{j_{d}} ω) |$ . Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate ...

On uniqueness for bounded channel flows of viscoelastic fluids

Marshall J. Leitman, Epifanio G. Virga (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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It was conjectured in [1] that there is at most one bounded channel flow for a viscoelastic fluid whose stress relaxation function $G$ is positive, integrable, and strictly convex. In this paper we prove the uniqueness of bounded channel flows, assuming $G$ to be non-negative, integrable, and convex, but different from a very specific piecewise linear function. Furthermore, whenever these hypotheses apply, the unbounded channel flows, if any, must grow in time faster than any polynomial. ...

Multiparameter ergodic Cesàro-α averages

A. L. Bernardis, R. Crescimbeni, C. Ferrari Freire (2015)

Colloquium Mathematicae

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Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^{p} (ν)$ , $T ₁, . . ., T_{k}$ , $n ̅ = (n ₁, . . ., n_{k}) \in ℕ^{k}$ and $α ̅ = (α ₁, . . ., α_{k})$ with $0 < α_{j} \leq 1$ , we define the ergodic Cesàro-α̅ averages $_{n ̅, α ̅} f = 1 / (\prod_{j = 1}^{k} A_{n_{j}}^{α_{j}}) \sum_{i_{k} = 0}^{n_{k}} \dots \sum_{i ₁ = 0}^{n ₁} \prod_{j = 1}^{k} A_{n_{j} - i_{j}}^{α_{j} - 1} T_{k}^{i_{k}} \dots T ₁^{i ₁} f$ . For these averages we prove the almost everywhere convergence on X and the convergence in the $L^{p} (ν)$ norm, when $n ₁, . . ., n_{k} \to \infty$ independently, for all $f \in L^{p} (d ν)$ with p > 1/α⁎ where $α ⁎ = m i n_{1 \leq j \leq k} α_{j}$ . In the limit case p = 1/α⁎, we prove that the averages $_{n ̅, α ̅} f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $Λ (1 / α ⁎, φ_{m - 1})$ with $φ ₘ (t) = t {(1 + l o g ⁺ t)}^{m}$ . To obtain the result in the limit case we need...

Moving averages

S. V. Butler, J. M. Rosenblatt (2008)

Colloquium Mathematicae

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In ergodic theory, certain sequences of averages $A_{k} f$ may not converge almost everywhere for all f ∈ L¹(X), but a sufficiently rapidly growing subsequence $A_{m_{k}} f$ of these averages will be well behaved for all f. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are $A_{k} f (x) = 1 / (2^{k}) \sum_{j = 4^{k} + 1}^{4^{k} + 2^{k}} f (T^{j} x)$ , then the subsequence $A_{k ²} f$ will not be pointwise good even on $L^{\infty}$ , but the subsequence $A_{2^{k}} f$ will be pointwise good on L¹. Understanding when the hyperexponential...

Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism

K. Frączek, M. Wysokińska (2008)

Colloquium Mathematicae

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We give a negative answer to a question put by Nadkarni: Let S be an ergodic, conservative and nonsingular automorphism on $(X ̃,_{X ̃}, m)$ . Consider the associated unitary operators on $L ² (X ̃,_{X ̃}, m)$ given by $U ̃_{S} f = \sqrt (d (m \circ S) / d m) \cdot (f \circ S)$ and $φ \cdot U ̃_{S}$ , where φ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that φ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.

The Morse minimal system is finitarily Kakutani equivalent to the binary odometer

Mrinal Kanti Roychowdhury, Daniel J. Rudolph (2008)

Fundamenta Mathematicae

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Two invertible dynamical systems (X,,μ,T) and (Y,,ν,S), where X and Y are Polish spaces and Borel probability spaces and T, S are measure preserving homeomorphisms of X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X₀ of X of measure one onto a subset Y₀ of Y of full measure such that (1) ${ϕ |}_{X ₀}$ is continuous in the relative topology on X₀ and $ϕ^{- 1} |_{Y ₀}$ is continuous in the relative topology on Y₀, (2) $ϕ (O r b_{T} (x)) = O r b_{S} (ϕ (x))$ for μ-a.e. x ∈ X. (X,,μ,T)...

Amenability and unique ergodicity of automorphism groups of Fraïssé structures

Andy Zucker (2014)

Fundamenta Mathematicae

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In this paper we consider those Fraïssé classes which admit companion classes in the sense of [KPT]. We find a necessary and sufficient condition for the automorphism group of the Fraïssé limit to be amenable and apply it to prove the non-amenability of the automorphism groups of the directed graph S(3) and the boron tree structure T. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering $G L (V_{\infty})$ , where $V_{\infty}$ is the countably...

Extensions of generic measure-preserving actions

Julien Melleray (2014)

Annales de l’institut Fourier

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We show that, whenever $Γ$ is a countable abelian group and $Δ$ is a finitely-generated subgroup of $Γ$ , a generic measure-preserving action of $Δ$ on a standard atomless probability space $(X, μ)$ extends to a free measure-preserving action of $Γ$ on $(X, μ)$ . This extends a result of Ageev, corresponding to the case when $Δ$ is infinite cyclic.

Unique Bernoulli $g$ -measures

Anders Johansson, Anders Öberg, Mark Pollicott (2012)

Journal of the European Mathematical Society

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We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a $g$ -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique $g$ -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the $g$ -measure.

Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^{d}$

Piotr Bugiel (1998)

Annales Polonici Mathematici

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Asymptotic properties of the sequences (a) ${P_{φ}^{j} g}_{j = 1}^{\infty}$ and (b) ${j^{- 1} \sum_{i = 0}^{j - 1} P ⁱ_{φ} g}_{j = 1}^{\infty}$ , where $P_{φ} : L ¹ \to L ¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov...

On Ordinary and Standard Lebesgue Measures on $ℝ^{\infty}$

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on $ℝ^{\infty}$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on $ℝ^{\infty}$ for α = (1,1,...).