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Branch of mathematics that studies dynamical systems
Krylov–Bogolyubov theorem Maximal ergodic theorem Ornstein isomorphism theorem Wiener–Wintner theorem Poincare recurrence theorem Kolmogorov extension theorem Kruskal
Ergodic_theory
The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics. Suppose that ( X , B , μ ) {\displaystyle (X,{\mathcal {B}}
Maximal_ergodic_theorem
restatement of the maximal ergodic theorem. If { f n } {\displaystyle \{f_{n}\}} is a martingale, we can define the martingale maximal function by f ∗ (
Maximal_function
Theorem in linear algebra
certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical
Perron–Frobenius_theorem
Area of mathematical analysis
maximal ergodic theorem is analogous to the Hardy–Littlewood maximal theorem: it bounds the measure of the set on which a supremum of ergodic averages is
Harmonic_analysis
theorem (dynamical systems) Kolmogorov–Arnold–Moser theorem (dynamical systems) Krylov–Bogolyubov theorem (dynamical systems) Maximal ergodic theorem
List_of_theorems
Romanian-American mathematician (1935–2025)
2, 2025. Retrieved May 7, 2025. Bourgain, Jean (1988). "On the maximal ergodic theorem for certain subsets of the integers". Israel Journal of Mathematics
Alexandra_Bellow
Solution to a stochastic differential equation
large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt
Diffusion_process
Identifies the commutant of a specific von Neumann algebra
In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the
Commutation theorem for traces
Commutation_theorem_for_traces
equation Chinese restaurant process Coupling (probability) Ergodic theory Maximal ergodic theorem Ergodic (adjective) Galton–Watson process Gauss–Markov process
List_of_probability_topics
Concept in statistics
uniformly distributed random phase. Where applicable, the central limit theorem dictates that at any point, the sum of these individual plane-wave contributions
Gaussian_random_field
Stochastic volatility model used in derivatives markets
large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt
SABR_volatility_model
Type of biased random walk on a graph
A maximal entropy random walk (MERW) is a popular type of biased random walk on a graph, in which transition probabilities are chosen accordingly to the
Maximal_entropy_random_walk
Representation of a type of random process
{\displaystyle X_{t}} is also a Gaussian process. In other cases, the central limit theorem indicates that X t {\displaystyle X_{t}} will be approximately normally
Autoregressive_model
population models Matrix t-distribution Mauchly's sphericity test Maximal ergodic theorem Maximal information coefficient Maximum a posteriori estimation Maximum
List_of_statistics_articles
large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt
Continuous-time stochastic process
Continuous-time_stochastic_process
Rate of separation of infinitesimally close trajectories
Lyapunov exponents will be the same for almost all starting points of an ergodic component of the dynamical system. To introduce Lyapunov exponent consider
Lyapunov_exponent
Berry–Tabor conjecture in quantum chaos Banach's problem – is there an ergodic system with simple Lebesgue spectrum? Birkhoff conjecture – if a billiard
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
bundle G / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space
Ergodic_flow
American mathematician
Foreman, Matthew; Weiss, Benjamin (2004). "An anti-classification theorem for ergodic measure-preserving transformations". Journal of the European Mathematical
Matthew_Foreman
Belgian mathematician (1954–2018)
1007/BF01388911. S2CID 123312114. Bourgain, Jean (1989). "Pointwise ergodic theorems for arithmetic sets". Publications Mathématiques de l'IHÉS. 69: 5–41
Jean_Bourgain
the Rokhlin lemma, or Kakutani–Rokhlin lemma is an important result in ergodic theory. It states that an aperiodic measure preserving dynamical system
Rokhlin_lemma
of the iterated logarithm / (S:R) Maximal ergodic theorem / (S:R) Op (statistics) / (S:R) Optional stopping theorem / (FS:R) Stationary process / (SU:R)
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem of H. Dye asserts that every ergodic nonsingular transformation
Markov_odometer
Numbers whose differences are not squares
Nagel, Rainer (2015), "20.5 The Furstenberg–Sárközy Theorem", Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, vol. 272, Cham
Square-difference-free_set
Mathematical model for neuron networks
large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt
Galves–Löcherbach_model
Argentine mathematician
from interpolation theory to Cauchy integrals on Lipschitz curves, from ergodic theory to inverse problems in electrical prospection. Calderón's work has
Alberto_Calderón
Left-invariant (or right-invariant) measure on locally compact topological group
group theory, representation theory, statistics, probability theory, and ergodic theory. Let ( G , ⋅ ) {\displaystyle (G,\cdot )} be a locally compact Hausdorff
Haar_measure
Power law growth of entropy of language or a stochastic process
perigraphic process is roughly an algorithmically random ergodic component of a non-ergodic process with a non-atomic invariant sigma-algebra. An example
Hilberg's_hypothesis
Lie group of complex numbers of unit modulus; topologically a circle
minimal and acts ergodically if and only if a {\displaystyle a} is irrational. It is uniquely ergodic in that case. One version of the ergodicity states that
Circle_group
Property of certain dynamical systems
often referred to as characterizing integrable systems: the existence of a maximal set of conserved quantities (the usual defining property of complete integrability)
Integrable_system
Lossless data compression algorithms
Shor. Formally, (Theorem 13.5.2 ). LZ78 is universal and entropic—If X {\textstyle X} is a binary source that is stationary and ergodic, then lim sup n
LZ77_and_LZ78
Type of mathematical method
and maximal size. Thus |S| ≤ n. By maximality, r is wandering for τ, a contradiction. Corollary. If an invertible transformation T acts ergodically but
Hopf_decomposition
Use of the second law of thermodynamics to distinguish past from future
many of the interesting cases are either ergodic or mixing, and it is strongly suspected that mixing and ergodicity somehow underlie the fundamental mechanism
Entropy_as_an_arrow_of_time
Branch of mathematics
about the dynamics of f on the support of the equilibrium measure: f is ergodic and, more strongly, mixing with respect to that measure, by Fornaess and
Complex_dynamics
French mathematician (born 1956)
MR 1734665. S2CID 106476. Zbl 0387.47038. Passty, Gregory B. (1979). "Ergodic convergence to a zero of the sum of monotone operators in Hilbert space"
Pierre-Louis_Lions
as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic
List_of_conjectures
Physical law for entropy and heat
statement was shown to be equivalent to the statement of Clausius. The ergodic hypothesis is also important for Ludwig Boltzmann's approach. It says that
Second_law_of_thermodynamics
Scientific study of digital information
For general sources and channels that are not necessarily stationary or ergodic, information-spectrum methods characterize coding limits using asymptotic
Information_theory
action on H 1 ( α ) {\displaystyle {\mathcal {H}}_{1}(\alpha )} that is ergodic with respect to ν 1 {\displaystyle \nu _{1}} . A half-translation surface
Translation_surface
Discrete subgroup in a locally compact topological group
homogeneous manifolds), in number theory (through arithmetic groups), in ergodic theory (through the study of homogeneous flows on the quotient spaces)
Lattice_(discrete_subgroup)
Concept in topology
compactification extend beyond partition regularity into density Ramsey theory and ergodic theory. While classical Ramsey theory asks which cell of a partition contains
Stone–Čech_compactification
Banach space with a compatible structure of a lattice
isomorphic to closed sublattices of L1([0,1]). The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces. Banach space –
Banach_lattice
Field of mathematics and science based on non-linear systems and initial conditions
These include, for example, measure-theoretical mixing (as discussed in ergodic theory) and properties of a K-system. A chaotic system may have sequences
Chaos_theory
Swedish mathematician
the solutions of this relativistic wave equation to dynamical systems, ergodic theory, and operator theory. This lead, in particular, to the theory of
Hakan_Hedenmalm
homogeneous. Moreover, the corresponding measure on X as per the previous theorem is ergodic. If X is a Borel G space and x ∈ X, then the fixed point subgroup
System_of_imprimitivity
in ergodic theory reduce to problems about automorphisms of abelian von Neumann algebras. In that regard, the following results are useful: Theorem. Suppose
Abelian_von_Neumann_algebra
Probability concept
S}q_{ij}k_{j}^{A}=1&{\text{ for }}i\notin A.\end{aligned}}} An instance of ergodic theory, the ergodic theorem states that for an irreducible aperiodic Markov chain, for
Discrete-time_Markov_chain
Chinese-American mathematician (born 1949)
of the fundamental group. The splitting theorem says that the splitting of the fundamental group as a maximally noncommutative direct product implies the
Shing-Tung_Yau
Graph of a dynamical system
Zbl 0264.46057. Vershik, A.M. (1985). "A theorem on the Markov periodic approximation in ergodic theory". Journal of Soviet Mathematics. 28 (5):
Bratteli_diagram
American mathematician
JSTOR 2043117. Dubins, Lester E.; Pitman, Jim (1979). "A Pointwise Ergodic Theorem for the Group of Rational Rotations". Transactions of the American
Lester_Dubins
South Korean American mathematician
Mathematical Society, vol. 26, 2013, pp. 511–562 with Amir Mohammadi: Ergodicity of unipotent flows and Kleinian groups, Journal of the American Mathematical
Hee_Oh
Principle in Bayesian statistics
statements do not imply that thermodynamical systems need not be shown to be ergodic to justify treatment as a statistical ensemble. In ordinary language, the
Principle_of_maximum_entropy
American mathematician
425: 71–86. doi:10.1515/crll.1992.425.71. Goldman, William M. (1997). "Ergodic theory on moduli spaces". Annals of Mathematics. 146 (3): 1–33. doi:10
William Goldman (mathematician)
William_Goldman_(mathematician)
Fifteen problems in mathematical physic
"Dynamics and Spectral Theory of Quasi-Periodic Schrödinger-type Operators". Ergodic Theory and Dynamical Systems. 37 (8): 2353–2393. arXiv:1503.05740. doi:10
Simon_problems
Form of differential geometry
Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. The systole of a compact metric space
Systolic_geometry
Type of mathematical group
called maximal Kleinian groups, and it is possible to give a complete classification in a given arithmetic commensurability class. A theorem of Margulis
Arithmetic_Fuchsian_group
Type of graph in mathematics and physics
almost all finite connected quantum graphs the probabilistic dynamics is ergodic and mixing, in other words chaotic. Quantum graphs embedded in two or three
Quantum_graph
Type of simplified mathematical model
Rademacher JD, Zagaris A (2016). Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Springer. ISBN 978-3-319-26883-5.
Coarse-grained_modeling
Romanian mathematician
Cornelia; Nowak, Piotr W. (2017). "Kazhdan projections, random walks and ergodic theorems". Journal für die reine und angewandte Mathematik. 2019 (754): 49–86
Cornelia_Druțu
years. ergodicity economics A research programme aimed at reworking the theoretical foundations of economics around the concept of ergodicity. The programme's
Glossary_of_economics
Monte Carlo algorithm
π ( x ) {\displaystyle \pi (x)} must be unique. This is guaranteed by ergodicity of the Markov process, which requires that every state must (1) be aperiodic—the
Metropolis–Hastings_algorithm
Mathematical measure space associated to a random walk
the identification problem". In Pollicott, Mark; Schmidt, Klaus (eds.). Ergodic theory of Zd actions (Warwick, 1993–1994). London Math. Soc. Lecture Note
Poisson_boundary
Exact solution for the Einstein field equations
gamma-ray bursts. The Kerr geometry exhibits many noteworthy features: the maximal analytic extension includes a sequence of asymptotically flat exterior
Kerr_metric
Model of quantum optics
affected by the presence of quantum scars display behaviors that break ergodicity. The Dicke model of Eq. 1 assumes that the cavity mode and the two-level
Dicke_model
Form of artificial neural network
Retrieved 2021-03-21. Süzen, Mehmet (29 September 2014). "Effective ergodicity in single-spin-flip dynamics". Physical Review E. 90 (3) 032141. arXiv:1405
Hopfield_network
Mathematical model of ferromagnetism in statistical mechanics
model. This system deserves interest in its own; particularly one has "non-ergodic" properties leading to strange relaxation behaviour. Much attention has
Ising_model
General relativity model near spacetime singularities
number of terms in the sum eq. 81 is large and according to general theorems of the ergodic theory the values of τs are distributed around τ s ¯ {\displaystyle
BKL_singularity
located anywhere on the plane. Under the assumption of point process ergodicity (satisfied when using homogeneous Poisson processes), the results for
Stochastic geometry models of wireless networks
Stochastic_geometry_models_of_wireless_networks
MAXIMAL ERGODIC-THEOREM
MAXIMAL ERGODIC-THEOREM
Boy/Male
Hindu, Indian
Praise
Girl/Female
Greek
Muse of erotic poetry.
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Tamil, Telugu, Traditional
A String of Pearls
Male
French
French form of Latin Maximus, MAXIME means "the greatest."Â
Boy/Male
Sikh
A king
Girl/Female
Indian, Indonesian, Italian
Gift of God; Periodic
Girl/Female
Hindu
Greatness
Girl/Female
Hindu
Full of jewel
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Fragrance
Girl/Female
Indian
Soft
Boy/Male
Hindu
Devoted, A promise to God
Boy/Male
American, Australian, Chinese, French, German, Greek, Latin, Swedish
Greatest
Boy/Male
British, English
Ermine; Ferret-like Mammal; Animal Name
Male
Russian
(МакÑим) Variant spelling of Russian Maksim, MAXIM means "the greatest." Compare with another form of Maxim.
Boy/Male
Celebrity, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sanskrit, Sikh, Tamil, Telugu, Traditional
King of the Earth; A King
Boy/Male
Hindu, Indian, Marathi
The Garland of Lord Vishnu
Boy/Male
Hindu, Indian, Punjabi, Sikh, Tamil
Great Speech
Boy/Male
Gujarati, Hindu, Indian
Rich; Maladar
Boy/Male
Hindu
Dignity, Power
Boy/Male
Latin
Greatest.
MAXIMAL ERGODIC-THEOREM
MAXIMAL ERGODIC-THEOREM
Boy/Male
Hindu, Indian, Jain
Lion; King of Forest
Boy/Male
Muslim
Fighter in the way of Allah, A warrior
Girl/Female
Hindu, Indian
Moon Light
Boy/Male
Arabic, Muslim
Servant of God
Boy/Male
African
Ghanian name given to the first child from a second husband.
Boy/Male
Hindu, Indian, Punjabi, Sikh
Judge of Character; King of God
Male
Portuguese
Portuguese form of Latin Eduardus, DUARTE means "guardian of prosperity."
Girl/Female
Indian
Great
Male
Italian
Italian and Spanish form of Roman Latin Severus, SEVERO means "stern."
Boy/Male
Arabic, Indian, Muslim
Name Derived from Self Sacrifice
MAXIMAL ERGODIC-THEOREM
MAXIMAL ERGODIC-THEOREM
MAXIMAL ERGODIC-THEOREM
MAXIMAL ERGODIC-THEOREM
MAXIMAL ERGODIC-THEOREM
a.
Pertaining to, or derived from, ergot; as, ergotic acid.
a.
Of or relating to animals; as, animal functions.
a.
Partaking of the nature both of vegetable and animal matter; -- a term sometimes applied to vegetable albumen and gluten, from their resemblance to similar animal products.
n.
An amorous composition or poem.
a.
Of or pertaining to rhodium; containing rhodium.
a.
Pertaining to, derived from, or designating, the highest oxygen acid (HIO/) of iodine.
a.
Conveying impressions from the surface of the body to the spinal cord; -- said of certain nerves. Opposed to exodic.
a.
Pertaining to the merely sentient part of a creature, as distinguished from the intellectual, rational, or spiritual part; as, the animal passions or appetites.
a.
Greatest in quantity or highest in degree attainable or attained; as, a maximum consumption of fuel; maximum pressure; maximum heat.
n.
A salt of periodic acid.
n.
An extract made from ergot.
a.
Conducting influences from the spinal cord outward; -- said of the motor or efferent nerves. Opposed to esodic.
v.
Of or pertaining to a husband; as, marital rights, duties, authority.
a.
Consisting of the flesh of animals; as, animal food.
a.
Pertaining to Argolis, a district in the Peloponnesus.
a.
Alt. of Periodical
pl.
of Maximum
a.
Belonging to the axis of the body; as, the axial skeleton; or to the axis of any appendage or organ; as, the axial bones.
n.
Erotic quality.
n.
An animal that suckles its young; a mammal.