David Fuller – Journalist, Reporter, Documentary Filmmaker & Writer

Yup.
http://davidfuller.tv/


~A.

Enjoy, I just found them....
https://lightwarriorslegion.com/description-and-interpretation-of-4-plates-by-manly-hall/

Description and interpretation of 4 Plates By Manly P. Hall

Chris A. Parker15 January 2020

In Occult by Chris A. ParkerLeave a Comment

Reading Time: 10 minutes

Manly P. Hall is one of my favorites writers. During my study of his writings, I found some very interesting plates, full of occult symbols, with description and interpretation by him. I thought that every occult student must have the opportunity to take advantage of these, and I decided to collect and present them in a blog post. Here are 4 of them.

• Plate No 1

  

  Image By "The All-seeing Eye" Magazine 1923

Click here to open the image in full size and in a separate window for better study

This plate is taken from the rare and unobtainable writings of Robert Fludd the medieval English alchemist and Rosicrucian who is said to have brought the teachings of C. R. C. from Germany and to have been closely connected with the early development of both Masonry and Rosicrucianism.

The plate represents the hierarchies of nature and its great lesson to the student of occult philosophy lies in the analogy between elements, chemicals, planets, gods and celestial hierarchies. The plate is divided into two grand divisions like the horoscope of astrology. That which is below the central horizontal line represents the inferior creation while that which is above symbolizes the superior creation. As the superior creation is the cause all world there is laid out in this chart the superphysical hierarchies and the various intelligent powers behind manifestation.

The upper half of the diagram is symbolical of the Masonic Lodge and the body of the enlightened Mason while the lower part symbolizes the unilluminated negation of being. In the concentric rings are placed the names of the Powers of the universe as they are found in the various sacred arts and sciences. The sacred Hebrew names and the Sephira of the Hebrew Qabballah are found in the spaces between A and B. The superphysical hierarchies of divine beings and the leading angels and rulers of the hierarchies pass through the sphere marked by the line of B. Under C. we have the astrological worlds and under D. the natural, chemical, alchemical, mineral and animal kingdoms laid out as they are found in nature.

In the outside rings beyond A., we find the primitive principles of creation with the part they play in the unfolding of a universe, an individual, or a protoplasmic cell. This is one of the most complicated of the alchemical plates and can never be satisfactorily explained until the individual has unfolded a very high degree of spiritual sight and insight. The passage of man through the spiritual worlds of nature and the twelvefold constitution of his own globe and chain is the result of conscious initiation which, until it takes place, conceals from man – because of his own consciousness limitation – the mysteries which are the heritage of the wise. There are really no mysteries in nature for those who have earned the right to know; neither is there anything concealed that shall not be revealed. But the only way that the unseen can be brought into conscious manifestation is when man removes the veils of limitation from his own eyes by growth and unfoldment. Thus these plates which we have been issuing in our magazine have a very great meaning but like the sacred scriptures of the Illumined are sealed forever from the ignorant by their own ignorance.

No mere intellectual power is capable of unveiling the divine mysteries. Only soul qualities, the highest of the spiritual reflective powers, the co-joining of spiritual reason and mystical intuition is capable of producing true illumination. The first step to the study of these plates is neither reading nor meditating but practical self-regeneration which will give the higher power in man an opportunity of expressing its own omnipotent knowledge. This plate contains the entire secret of spiritual rulership and analogy; but no more may be said about it than that each individual must file from his own organisms the key which shall unlock its mysteries for the wise designed these things for the use of the wise and the price of understanding the words of the Illuminated is to become illuminated yourself. This is done when the light of spirit shines forth to bring out the colors on the printed page through the regenerated lantern of the philosopher-his own sevenfold body organism. (Further reading: Occult Q&A By Manly P. Hall)

• Plate No 2

Image By "The All-seeing Eye" Magazine 1923

Click here to open the image in full size and in a separate window for better study

This plate is taken from the rare and unobtainable work of Kunrath, the great alchemist, represents symbolically human regeneration and 1s also the key to the Philosopher's Stone.

As before, the translating of it shall be left to you, because it is only in that way you can really learn its message. But we will briefly consider some of its most important symbols: The figure rising out of the globe symbolizes spirit rising out of matter and consciousness freeing itself from the encircling and enslaving bonds of form. The two-headed figure represents the Hermetic union and the creation of Azoth the Philosopher's Stone. In this plate we have the answer to the problem of soul-mates as only the ancient alchemist could explain it, for the male-female creature here shown symbolizes the occult constitution of man who is the male-female creation. The male figure has the sun halo or the positive ray while the female figure has the moon crown or the mother ray, representing spirit and matter, which matter being regenerated becomes the soul or bride of spirit.

This figure rises out of the globe of elements and from the heads arises a wondrous bird with the sign of Leo around its head. This blackened bird represents the unknowable secret of the phoenix or the bird of eternal life that is born out of the union of the sun and moon in the brain of man. Its tail, which is filled with eyes, represents the unfolded sense centers of human consciousness while the great circle containing all the other symbols is made to represent nature within whose protecting aura all growth is carried on.

The fire of the philosopher which rises upward and partly surrounds the central globe is the purification process in which the flame in the lower centers of the body rises upward and awakens Kundalini, the spinal spirit-fire in man, which is asleep in the egg of Brahma located in the solar plexus. This passing upward creates the figure with the two heads for these faces undoubtedly represent the pituitary body and the pineal gland which are the positive and negative poles of the spinal canal fire. In India the god-man lshwari is shown as a male-female Diety and in the ancient languages the name of God signified that He was also a male-female Divinity, for He is not only the Creator but the Creation. In a similar way man, following in the footsteps of God, is slowly arousing the latent qualities within himself and building to the day when he too shall be both creator and the creation.

The entire diagram is symbolical of the evolution of the human soul and spirit. Starting from the top downward it is involution; working from the bottom upward it is evolution. Two streams pour from the breasts of the creature and these represent the outpourings of fire and water or salt and sulphur which are two of the three elements of perfection while mercury forms the third element. The band around the neck of the figure, which unites the heads, is the wedding ring of modem theology for it ties or unites as a band of spiritual gold the two extremes of human life.

The upright triangle above, pointing up to the Sacred Name, is once more a symbol of human regeneration. Taking the plate generally it refers to the cosmic scheme of things and later the individual scheme of things. The reading tells of how through the union of the universal Earth Mother and Fiery and Airy Father there is created a wondrous stone which is the answer to all the problems of life.

The student recognizes that the union of the spiritual elements within himself will turn him allegorically into a two-headed creature male-female and self-reproductive through the positive pole of the brain.

Check out our recommendations at "Occult Bookshelf" and many free resources at our "Free Library"

• Plate No 3

Image By "The All-seeing Eye" Magazine 1923

Click here to open the image in full size and in a separate window for better study

This picture is a companion piece to the previous one and is taken from the writings of Henry Kunrath the great medieval alchemist and kabbalist and supposed member of the ancient Rosicrucian brotherhood.

The figure represents Hercules the androgenous Man Christ strangling the serpent of evil and exalting the serpent of wisdom. At the top of the drawing is represented the divine name Jehovah, the male-female third aspect of the Logos and the God of material growth. From him pour out the archetypes of the ten Sephira or the original ten signs of the Zodiac and powers of cosmic impetus.

All things first exist in archetype or cosmic plan before they are concreted in matter. Material substance is poured like molten metal into the superphysical mold in the creation of bodies. In this diagram is concealed the Sacred Name. The upright triangle is an A. and means Adonai the Fireflame. The broken circle of the border forms the U. of the Universe, while the cube of matter crystallized within it is the M. of the Manus worlds. Again the drawing is that of man with the spiritual triangle, the mental hook or U. and the bodily cube of M.

The ability to alchemically combine these three superior elements is the key to occultism and the philosopher's stone. The book sealed with seven seals represents wisdom unlocked by the seven keys of' the occult schools each one of which is a musical note, a color, a sound, an internal body chrakra, and a cosmic God. The passing of the consciousness of man through the seven worlds of nature constitutes the breaking of tile seals in which he learns to connect himself consciously with the plan of his own being. By means of this slow evolutionary development, man ascends the endless spiral of the Wisdom School.

The little figure in the center has two heads like the previous plate. These two represent the bride and the groom in the spiritual marriage and the completion of the eternal romance in the soul of man. The male head represents the spirit, the female head the soul; in other words, the sun and moon or the blood of the red lion and the gluten of the white eagle.

The ancients said that their immortal stone was made of moonstone and gold in certain proportions, absorbed by the universal vinegar of the prepared mercury. This alchemical process of transmuting the base qualities and perversions of power into spiritual attributes, accomplishments, and qualities is the key to the sacred wisdom of the ancients. This is attained through the upright triangle, symbolizing the upturned flame of the spinal fire which performs in the marrow of the bone the mystery of alchemy.

Among the Masons, Mahabone has been given as a substitute for the unspeakable and unknowable Word. Mahabone means the marrow in the bone and the secret of alchemy lies in the marrow of the spinal canal which contains within itself all mystery and is the staff of Brahma which supports the universe and all that it contains. (Further reading: Vocabulary of Occult Terms (A to J))

• Plate No 4

Image By "The All-seeing Eye" Magazine 1923

Click here to open the image in full size and in a separate window for better study

Thiplate isis the frontispiece of a rare and unobtainable work by Robert Fludd, the great English freemason, alchemist and Rosicrucian. The original folio was printed in 1619 in Latin and is really two books in one. The first book deals with the metaphysical creation of the heavens and earth and is a work of a student of Rosicrucian lore. It is now generally admitted that Robert Fludd was connected with that sacred order. He is said to be the first English exponent of cosmological alchemy and the philosophy of the Phoenicians and Chaldeans and is known all over the world as one of the deepest occultists of any generation known to man.

Technically the plate is astrological, dealing especially with the planetary centers in the human body and also the centers of the twelve signs of the zodiac. You can easily trace the position of the twelve signs by following the dotted line on the human figure, starting with Aries which governs the head and ending with Pisces which governs the feet.

The power which is turning the wheel of eternity and unwinding the cord of human destiny is the threefold beast which has since become a part of the Royal Arch banner of freemasonry. This creature is the most outstanding feature of the plate. It represents the three great principles of nature, manifesting through the three grand divisions or kingdoms of his own body.

The feet belong to the animal world, the human body belongs to the human world, while the wings belong to the celestial or divine world. The wings represent the creative power of God the Father, the human body represents the preservative power of God the Son, while the legs and feet represent the procreative and disintegrative power of God the Holy Spirit. On his head, the creature carries an hourglass which shows the passing of time and illustrates the principle that the spirit of Time is eternally unwinding the cord, which unwinding causes the universe to twist on its central axis.

The whole diagram is surrounded by clouds which represent Chaos and the great sphere is Cosmos in Chaos. The human body represents the five-pointed star of Masonry and also shows the position of the various centers of the human body in relation to the threefold world of nature. This is the microcosmic and macrocosmic man; in other words, the evolving human consciousness and also the cosmic consciousness of nature.

The twelve concentric circles of the outer sphere represent the twelve spiritual hierarchies or the worlds of the external heaven. The seven spheres in the secondary circle represent the home of the seven Elohim or planetary deities while the three inner worlds represent water, fire, and air, and the solid globe behind the figure the principle of earth.

The five points where the human body touches the sphere of the seven planes represent the sense perceptions of the human consciousness, while the little figure of Saturn over the head of the figure is the key of source. Above the figure is the terrestrial sun and moon, while still higher are the celestial sun and moon, much greater and more brilliant. The heart and mind are the sun and moon of the human system and in their union lies the power of an Initiate. Time is turning the Wheel of Life round and round; sometimes man stands upright as he is shown here, later he is inverted and assumes the position which you can study by inverting the picture. And this endless going round and round, first up and then down, is the Wheel of Life to which the threefold deity of concrete creation chains the spirit of man.

Only when he releases himself from the wheel of creation is he capable of releasing himself from the wheel of destruction, for as the ancients said, "Sure is death for the living, and sure is birth for the dead." The wisdom of the ancients lay not in combatting the principles of nature but in freeing themselves by their knowledge and understanding from the Wheel of Life and Death. This is the tenth card of the Taro, the Wheel of Fortune.

Finally, as the author suggests, "Save these pictures for you will find it nearly impossible to get them again, and while you may not understand them now, as time goes on you will be grateful that you possess them."

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Source: "The All-seeing Eye" Magazine By Manly P. Hall
Feature Image By "The All-seeing Eye" Magazine 1923

Check out our recommendations at "Occult Bookshelf" and many free resources at our "Free Library"

~A.

Collectively, the minds that are occupied in pursuing these individual and related theories on theoretical and potentially applied concepts appear wholly unrelated to Society's Bubble. Can the two support each other?
http://bactra.org/notebooks/ergodic-theory.html

Ergodic Theory

22 Aug 2019 17:05

---

A measure on a mathematical space is a way of assigning weights to different parts of the space; volume is a measure on ordinary three-dimensional Euclidean space. Probability distributions are measures, such that the largest measure of any set is 1 (and some other restrictions). Suppose we're interested in a dynamical system --- a transformation that maps a space into itself. The set of points we get from applying the transformation repeatedly to a point is called its trajectory or orbit. Some dynamical systems are "measure preserving", meaning that the measure of a set is always the same as the measure of the set of points which map to it. (In symbols, using

$T$

for the map and

$P$

for the probability measure,

$P(T^{-1}(A))=P(A)$

for any measureable set

$A$

.) Some sets may be invariant: they are the same as their images. An ergodic dynamical system is one in which, with respect to some probability distribution, all invariant sets either have measure 0 or measure 1. (Sometimes non-ergodic systems can be decomposed into a number of components, each of which is separately ergodic.) The dynamics need not be deterministic; in particular, irreducible Markov chains with finite state spaces are ergodic processes, since they have a unique invariant distribution over the states. (In the Markov chain case, each of the ergodic components corresponds to an irreducible sub-space.)

Ergodicity is important because of the following theorem (due to von Neumann, and then improved substantially by Birkhoff, in the 1930s). If we take any well-behaved (integrable) function of our space, pick a point in the space at random (according to the ergodic distribution) and calculate the average of the function along the point's orbit, the time-average. Then, with probability 1, in the limit as the time goes to infinity, (1) the time-average converges to a limit and (2) that limit is equal to the weighted average of the value of the function at all points in the space (with the weights given by the same distribution), the space-average. The orbit of almost any point you please will in some sense look like the whole of the state space.

(Symbolically, write

$x$

for a point in the state space,

$f$

for the function we're averaging, and

$T$

and

$P$

for the map and the probability measure as before. The space-average,

$\overline{f}=\int f(x)P(dx)$

. The time-average starting from

$x$

,

${⟨f⟩}_x=\underset{n\to \mathrm{\infty }}{lim}(1/n)\sum _{i=0}^{n}f(T^i(x))$

. The ergodic theorem asserts that if

$f$

is integrable and

$T$

is ergodic with respect to

$P$

, then

${⟨f⟩}_x$

exists, and

$P\left\{x:{⟨f⟩}_x=\overline{f}\right\}=1$

. --- A similar result holds for continuous-time dynamical systems, where we replace the summation in the time average with an integral.)

This is an extremely important property for statistical mechanics. In fact, the founder of statistical mechanics, Ludwig Boltzmann, coined "ergodic" as the name for a stronger but related property: starting from a random point in state space, orbits will typically pass through every point in state space. It is easy to show (with set theory) that this isn't doable, so people appealled to a weaker property which was for a time known as "quasi-ergodicity": a typical trajectory will pass arbitrarily close to every point in phase space. Finally it became clear that only the modern ergodic property is needed. To see the relation, consider the function, call it

$I_A$

, which is 1 on a certain set

$A$

, and 0 elsewhere. (This is the "indicator function" of

$A$

.) The time-average of

$I_A$

is the fraction of time that the orbit spends in

$A$

. The space-average of

$I_A$

is the probability that a randomly picked point is in

$A$

. Since the two averages are almost always equal, almost all trajectories end up covering the state space in the same way.

One way of thinking about the classical ergodic theorem is that it's a version of the law of large numbers --- it tells us that a sufficiently large sample (i.e., an average over a long time) is representative of the whole population (i.e., the space average). One thing I'd like to know more about than I do is ergodic equivalents of the central limit theorem, which say how big the sampling fluctuations are, and how they're distributed. The other thing I want to know about is the rate of convergence in the ergodic theorem --- how long must I wait before my time average is within a certain margin of probable error of the state average. Here I do know a bit more of the relevant literature, from large deviations theory.

Again in symbols: Let's write

${⟨f⟩}_{x,n}$

for the time-average of

$f$

, starting from

$x$

, taken over

$n$

steps. Then a central-limit theorem result would say that (for example)

$\frac{{⟨f⟩}_{X,n}-\overline{f}}{{\sigma }^{2}r(n)}$

converges in distribution to a Gaussian with mean zero and variance one, where

${\sigma }^2$

is the (space-averaged) variance of

$f$

and

$r(n)$

is some positive, increasing function of

$n$

. This would be weak convergence of the time averages to the space averages, and

$r(n)$

would give the rate. (In the usual IID case,

$r(n)=\sqrt{n}$

.) Somewhat stronger would be a convegence in probability result, giving us a function

$N(\epsilon ,\delta )$

such that

$P\left\{x:\left|{⟨f⟩}_{x,n}-\overline{f}\right|\ge \epsilon \right\}\le \delta$

if

$n\ge N(\epsilon ,\delta )$

. Proving many of these results requires stronger assumptions than proving ergodicity does --- for instance, mixing properties.

These issues are part of a more general question about how to do statistical inference for stochastic processes, a.k.a. time-series analysis. I am especially interested in statistical learning theory in this setting, which is in part about ensuring that the ergodic theorem holds uniformly across classes of functions. Very strong results have recently been achieved on that front by Adams and Nobel (links below).

Another thing I'd like to understand, but don't have time to explain here, are Pinsker sigma-algebras.

See also: Deviation Inequalities; Dynamical Systems and Chaos; Empirical Process Theory; Information Theory; Mixing and Weak Dependence of Stochastic Processes; Nonequilibrium Statistical Mechanics; Probability Theory; Recurrence Times of Stochastic Processes; Stochastic Processes; Symbolic Dynamics; Time Series; Universal Prediction Algorithms

Recommended, synoptic:
• Peter Billingsley, Ergodic Theory and Information
• Robert M. Gray, Probability, Random Processes, and Ergodic Properties [Full-text online]
• A. I. Khinchin, Mathematical Foundations of Statistical Mechanics [Proves the von Neumann-Birkhoff ergodic theorem in detail]
• Mark Kac, Probability and Related Topics in Physical Science
• Andrzej Lasota and Michael C. Mackey, Chaos, Fractals and Noise: Stochastic Aspects of Dynamics
• Norbert Wiener, Cybernetics: Or Control and Communication in the Animal and the Machine

Recommended, close-up:
• Terrence M. Adams and Andrew B. Nobel [See comments under Statistical Learning with Dependent Data]
  "Uniform convergence of Vapnik-Chervonenkis classes under ergodic sampling", Annals of Probability 38 (2010): 1345--1367, arxiv:1010.3162
  • "The Gap Dimension and Uniform Laws of Large Numbers for Ergodic Processes", arxiv:1007.2964
• Itai Benjamini, Nicolas Curien, "Ergodic Theory on Stationary Random Graphs", arxiv:1011.2526
• Leo Breiman, "The Individual Ergodic Theorem of Information Theory", Annals of Mathematical Statistics 28 (1957): 809--811 [This includes a proof of an interesting and useful result about ergodic convergence of sequences of functions]
• Richard C. Bradley, "Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions", Probability Surveys 2 (2005): 107--144, arxiv:math/0511078
• J.-R. Chazottes and R. Leplaideur, "Birkhoff averages of Poincare cycles for Axiom-A diffeomorphisms," math.DS/0312291
• Jérôme Dedecker, Paul Doukhan, Gabriel Lang, José Rafael León R., Sana Louhichi and Clémentine Prieur, Weak Dependence: With Examples and Applications [Mini-review]
• Paul Doukhan, Mixing: Properties and Examples
• E. B. Dynkin, "Sufficient statistics and extreme points", Annals of Probability 6 (1978): 705--730 ["The connection between ergodic decompositions and sufficient statistics is explored in an elegant paper by DYNKIN" --- Kallenberg, Foundations of Modern Probability, p. 577]
• Jean-Pierre Eckmann and David Ruelle, "Ergodic Theory of Chaos and Strange Attractors," Reviews of Modern Physics 57 (1985): 617--656
• Roberto Fernández and Grégory Maillard, "Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties", Journal of Statistical Physics 118 (2005): 555--588
• Harry Furstenberg, Stationary Processes and Prediction Theory
• Stefano Galatolo, Mathieu Hoyrup, and Cristóbal Rojas, "Effective symbolic dynamics, random points, statistical behavior, complexity and entropy", arxiv:0801.0209 [All, not almost all, Martin-Löf points are statistically typical.]
• Weihong Huang, "On the long-run average growth rate of chaotic systems", Chaos 14 (2004): 38--47 [An amusing demonstration that positive-valued ergodic processes will seem to always have a positive long-run growth rate, even though they're stationary!]
• Michael Keane and Karl Petersen, "Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem", math.DS/0608251 [This is a lovely little four-page paper, and the simplest proof by far that I've seen, but they do rely rather heavily on the reader being familiar with facts about time averages, invariant functions, etc. Still, I should definitely teach this in my class.]
• J. F. C. Kingman, "Subadditive Ergodic Theory", Annals of Probability 1 (1973): 883--909 [with discussion and reply]
• Aryeh (Leo) Kontorovich, "Metric and Mixing Sufficient Conditions for Concentration of Measure", math.PR/0610427 [See weblog comments]
• Aryeh Kontorovich and Anthony Brockwell, "A Strong Law of Large Numbers for Strongly Mixing Processes", arxiv:0807.4665
• Michael C. Mackey, Time's Arrow: The Origins of Thermodynamic Behavior
• Florence Merlevède, Magda Peligrad, Sergey Utev, "Recent advances in invariance principles for stationary sequences", math.PR/0601315 = Probability Surveys 3 (2006): 1--36 [Can I just say how much I hate calling the functional central limit theorem "the invariance principle"?]
• Mehryar Mohri and Afshin Rostamizadeh, "Stability Bound for Stationary Phi-mixing and Beta-mixing Processes", arxiv:0811.1629 ["Stability" is the property of a learning algorithm that changing a single observation in the training set leads to only small changes in predictions on the test set. This paper shows that stable learning algorithms continue to perform well with dependent data, provided the data are either phi mixing or beta mixing.]
• Andrew Nobel and Amir Dembo, "A Note on Uniform Laws of Averages for Dependent Processes", Statistics and Probability Letters 17 (1993): 169--172 [PDF preprint via Prof. Nobel. See comments under Statistical Learning with Dependent Data]
• Donald Ornstein and Benjamin Weiss, "How Sampling Reveals a Process", Annals of Probability 18 (1990): 905--930 [Some comments under Universal Prediction.]
• Goran Peskir, From Uniform Laws of Large Numbers to Uniform Ergodic Theorems [PDF reprint via Prof. Peskir]
• Murray Rosenblatt, "A Central Limit Theorem and a Strong Mixing Condition", Proceedings of the National Academy of Sciences (USA) 42 (1956): 43--47 [The root from which much subsequent ergodic theory has sprung. PDF reprint]
• Daniil Ryabko and Boris Ryabko, "Testing Statistical Hypotheses About Ergodic Processes", arxiv:0804.0510
• Nobusumi Sagara, "Nonparametric maximum-likelihood estimation of probability measures: existence and consistency", Journal of Statistical Planning and Inference 133 (2005): 249--271 ["This paper formulates the nonparametric maximum-likelihood estimation of probability measures and generalizes the consistency result on the maximum-likelihood estimator (MLE). We drop the independent assumption on the underlying stochastic process and replace it with the assumption that the stochastic process is stationary and ergodic. The present proof employs Birkhoff's ergodic theorem and the martingale convergence theorem. The main result is applied to the parametric and nonparametric maximum-likelihood estimation of density functions." Very cool.]
• Yi Shen and Tony S. Wirjanto, "Stationarity Tests for Time Series -- What Are We Really Testing?", arxiv:1505.01163
• Paul C. Shields, The Ergodic Theory of Discrete Sample Paths [Well-written modern text, extremely strong on connections to information theory and coding. I haven't gotten through the last chapter, however.]
• Leslie Sklar, Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics [Good discussion of ergodic results in several places]
• J. Michael Steele, "Kingman's subadditive ergodic theorem", Annales de L'Institut Henri Poincare B 25 (1989): 93--98
• Marta Tyran-Kaminska, "An Invariance Principle for Maps with Polynomial Decay of Correlations", math.DS/0408185 = Communications in Mathematical Physics 260 (2005): 1--15 ["We give a general method of deriving statistical limit theorems, such as the central limit theorem and its functional version, in the setting of ergodic measure preserving transformations. This method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations."]
• Mathukumalli Vidyasagar, A Theory of Learning and Generalization: With Applications to Neural Networks and Control Systems [Has a very nice discussion of when the uniform laws of large numbers of statistical learning theory transfer from the usual IID setting to dependent processes, becoming uniform ergodic theorems. (Sufficient conditions include things like beta-mixing, but necessary and sufficient conditions seem to still be unknown.) Mini-review]
• Benjamin Weiss, Single Orbit Dynamics
• Wei Biao Wu, "Nonlinear system theory: Another look at dependence", Proceedings of the National Academy of Sciences 102 (2005): 14150--14154 ["we introduce [new] dependence measures for stationary causal processes. Our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems. The proposed dependence measures provide a natural framework for a limit theory for stationary processes. In particular, under conditions with quite simple forms, we present limit theorems for partial sums, empirical processes, and kernel density estimates. The conditions are mild and easily verifiable because they are directly related to the data-generating mechanisms."]

Recommended, historical:
• G. I. Taylor, "Diffusion by Continuous Movements", Proceedings of the London Mathematical Society, series 2, volume 20 (1922), pp. 196--212 [This seems to be the ultimate source for the very easy, but very practical, mean-square ergodic theorem that I've sometimes called the "world's simplest ergodic theorem" (link below)]

Modesty forbids me to recommend:
• chs. 5 and 22--27 of Almost None of the Theory of Stochastic Processes
• "The World's Simplest Ergodic Theorem"

To read:
• Jon Aaronson, An Introduction to Infinite Ergodic Theory
• Jon Aaronson and Tom Meyerovitch, "Absolutely continuous, invariant measures for dissipative, ergodic transformations", math.DS/0509093
• Nabil Al-Najjar, Eran Shmaya, "Learning the ergodic decomposition", arxiv:1406.6670
• Jose M. Amigo, Matthew B. Kennel and Ljupco Kocarev, "The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems", nlin.CD/0503044
• Abdelhamid Amroun, "Equilibrium states for smooth maps", arxiv:1004.2577
• Ari Arapostathis, Vivek S. Borkar, Mrinal K. Ghosh, Ergodic Control of Diffusion Processes
• Vitor Araujo, "Semicontinuity of entropy, existence of equilibrium states and of physical measures", math.DS/0410099
• L. Arnold, Random Dynamical Systems
• V. I. Arnol'd and A. Avez, Ergodic Problems of Classical Mechanics
• Jeremy Avigad, Philipp Gerhardy and Henry Towsner, "Local stability of ergodic averages", arxiv:0706.1512 [Computing bounds on the rate of convergence in the ergodic theorems; sounds cool. Thanks to Gustavo Lacerda for the pointer.]
• Jeremy Avigad, Jason Rute, "Oscillation and the mean ergodic theorem for uniformly convex Banach spaces", Ergodic Theory and Dynamical Systems 35 (2014): 1009--1027, arxiv:1203.4124
• Massimiliano Badino, "The Foundational Role of Ergodic Theory", phil-sci/2277
• Dominique Bakry, Patrick Cattiaux, Arnaud Guillin, "Rate of Convergence for ergodic continuous Markov processes: Lyapunov versus Poincare", math.PR/0703355
• Viviane Baladi, Positive Transfer Operators and Decay of Correlations
• Joseph Berkovitz, Roman Frigg and Fred Kronz, "The Ergodic Hierarchy, Randomness and Hamiltonian Chaos", phil-sci/2927
• A. A. Borovkov, Ergodicity and Stability of Stochastic Processes
• Philippe Chassaing, Jean Mairesse, "A non-ergodic PCA with a unique invariant measure", arxiv:1009.0143 [Here "PCA"="probabilistic cellular automaton"]
• J.-R. Chazottes and P. Collet, "Almost-sure central limit theorems and the Erdös-Rényi law for expanding maps of the interval", Ergodic Theory and Dynamical Systems 25 (2005): 419--41
• J.-R. Chazottes, P. Collet and B. Schmitt, "Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems", Nonlinearity 18 (2005): 2323--2340, math.DS/0412166
• J.-R. Chazottes, P. Collet and B. Schmitt, "Statistical Consequences of Devroye Inequality for Processes. Applications to a Class of Non-Uniformly Hyperbolic Dynamical Systems", Nonlinearity 18 (2005): 2341--2364, math.DS/0412167
• J.-R. Chazottes and G. Gouezel, "On almost-sure versions of classical limit theorems for dynamical systems", math.DS/0601388 [arguing in support of the idea that "whenever we can prove a limit theorem in the classical sense for a dynamical system, we can prove a suitable almost-sure version based on an empirical measure with log-average".]
• J.-R. Chazottes, G. Keller, "Pressure and Equilibrium States in Ergodic Theory", arxiv:0804.2562
• J.-R. Chazottes and F. Redig, "Testing the irreversibility of a Gibbsian process via hitting and return times", math-ph/0503071 = Nonlinearity 18 (2005): 2477--2489
• Mu-Fa Chen
  • Eigenvalues, Inequalities, and Ergodic Theory
  • "Ergodic convergence rates of Markov processes--eigenvalues, inequalities and ergodic theory", math.PR/0304367
• Xia Chen, Limit Theorems for Functionals of Ergodic Markov Chains with General State Space
• Geon Ho Choe, Computational Ergodic Theory
• Christophe Cuny, "Pointwise ergodic theorems with rate and application to limit theorems for stationary processes", arxiv:0904.0185
• Christophe Cuny, Dalibor Volny, "A quenched invariance principle for stationary processes", arxiv:1202.4875
• Lukasz Debowski, "Variable-Length Coding of Two-sided Asymptotically Mean Stationary Measures", Journal of Theoretical Probability 23 (2009): 237--256
• J. Dedecker, F. Merlevède, M. Peligrad, "Invariance principles for linear processes. Application to isotonic regression", arxiv:0903.1951 ["maximal inequalities and study the functional central limit theorem for the partial sums of linear processes generated by dependent innovations. Due to the general weights these processes can exhibit long range dependence and the limiting distribution is a fractional Brownian motion"]
• Thierry De La Rue, "An introduction to joinings in ergodic theory", math.DS/0507429 = Discrete and Continuous Dynamical Systems 15 (2006): 121--142
• Manfred Denker, Mikhail Gordin, "On conditional central limit theorems for stationary processes", pp. 133--151 in Krishna Athreya, Mukul Majumdar, Madan Puri, and Edward Waymire (eds.), Probability, statistics and their applications: papers in honor of Rabi Bhattacharya [Free online]
• G. B. DiMasi and L. Stettner, "Ergodicity of hidden Markov models", Mathematics of Control, Signals, and Systems 17 (2005): 269--296
• Ian Domowitz and Mahmoud El-Gamal, "A Consistent Nonparametric Test of Ergodicity for Time Series with Applications", Journal of Econometrics 102 (2001): 365--398 [SSRN. Having read about 2/3 of this, I completely fail to see how they actually overcome the problem that any one sample path is always confined to a single ergodic component.]
• Tomasz Downarowicz, Entropy in Dynamical Systems
• Aryeh Dvoretzky, "Asymptotic normality for sums of dependent random variables", Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, 513--535
• Martin Dyer, Leslie Ann Goldberg, Mark Jerrum, Russell Martin, "Markov chain comparison", math.PR/0410331 [i.e., comparison theorems for mixing times]
• Jean-Pierre Eckmann and Itamar Procaccia, "Invariant Measures in Generic Dynamical Systems", chao-dyn/9708021 [Abstract: "Irreversible thermodynamics of simple fluids have been connected recently to the theory of dynamical systems and some interesting assumptions have been made about the nature of the associated invariant measures. We show that the tests of the validity of these assumptions are insufficient by exhibiting observables that are incorrectly sampled with the proposed invariant measures. Only observables belonging to the 'high temperature phase' of the thermodynamic formalism are insensitive to the sampling methods. We outline methods that are free of these deficiencies." I read this when it came out, but I don't think I understood it then.]
• T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory
• Mohamed El Machkouri, Dalibor Volny, Wei Biao Wu, "A central limit theorem for stationary random fields", arxiv:1109.0838
• Bernhold Fiedler (ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems
• Nikos Frantzikinakis, Randall McCutcheon, "Ergodic Theoy: Recurrence", arxiv:0705.0033
• Gary Froyland, "Statistical optimal almost-invariant sets", Physica D 200 (2005): 205--219 [Partitioning state space into nearly separated components.]
• Harry Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory
• Stefano Galatolo, Mathieu Hoyrup, Cristóbal Rojas
  • "Dynamics and abstract computability: computing invariant measures", arxiv:0903.2385
  • "Dynamical systems, simulation, abstract computation", arxiv:1101.0833
  • "A constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties", arxiv:0711.1478
• Gan Shixin, "Almost sure convergence for
  $\tilde{\rho }$
  -mixing random variable sequences", Statistics and Probability Letters 67 (2004): 289--298
• E. Glasner and B. Weiss, "On the interplay between measurable and topological dynamics", math.DS/0408328
• Beniamin Goldys and Bohdan Maslowski, "Uniform exponential ergodicity of stochastic dissipative systems," math.PR/0111143
• Alexander Gorodnik and Amos Nevo, Ergodic Theory of Lattice Subgroups
• Sebastien Gouezel, "Almost sure invariance principle for dynamical systems by spectral methods", Annals of Probability 38 (2010): 1639--1671
• M. Hairer, "Ergodic properties of a class of non-Markovian processes", arxiv:0708.3338
• P. Halmos, Ergodic Theory
• Bruce E. Hansen, "Strong Laws for Dependent Heterogeneous Processes", Econometric Theory 7 (1991): 213--221 [PDF reprint via Prof. Hansen]
• Nicolai T. A. Haydn, "The Central Limit Theorem for uniformly strong mixing measures", arxiv:0903.1325
• Nicolai Haydn, Y. Lacroix and Sandro Vaienti, "Hitting and return times in ergodic dynamical systems", math.DS/0410384 = Annals of Probability 33 (2005): 2043--2050
• Nicolai Haydn and Sandro Vaienti, "Fluctuations of the Metric Entropy for Mixing Measures", Stochastics and Dynamics 4 (2004): 595--627
• Michael Hochman, "Upcrossing Inequalities for Stationary Sequences and Applications to Entropy and Complexity", arxiv:math.DS/0608311 [where "complexity" = algorithmic information content]
• K. Horbacz, M. ÅšlÄ™czka, "Law Of Large Numbers For Random Dynamical Systems", arxiv:1304.6863
• Bernard Host, "Convergence of multiple ergodic averages", math.DS/0606362 ["We study the mean convergence of multiple ergodic averages, that is, averages of a product of functions taken at different times."]
• Steven Kalikow and Randall McCutcheon, An Outline of Ergodic Theory
• Gerhard Keller, Equilibrium States in Ergodic Theory
• Gerhard Keller and Carlangelo Liverani, "Uniqueness of the SRB Measure for Piecewise Expanding Weakly Coupled Map Lattices in Any Dimension", Communications in Mathematical Physics 262 (2006): 33--50
• Yuri Kifer, "From Pet to Split", arxiv:0809.4106
• U. Kregnel, Ergodic Theorems
• Anna Kuczmaszewska, "The strong law of large numbers for dependent random variables", Statistics and Probability Letters 73 (2005): 305--314
• S. N. Lahiri, "Asymptotic expansions for sums of block-variables under weak dependence", arxiv:math/0606739
• Lin Zhengyan and Lu Chuanrong, Limit Theory for Mixing Dependent Random Variables
• Pei-Dong Liu and Min Qian, Smooth Ergodic Theory of Random Dynamical Systems
• Martial Longla, Magda Peligrad, "Some Aspects of Modeling Dependence in Copula-based Markov chains", arxiv:1107.1794
• Dasha Loukianova, Oleg Loukianov, Eva Loecherbach, "Polynomial bounds in the Ergodic Theorem for positive recurrent one-dimensional diffusions and integrability of hitting times", arxiv:0903.2405 [non-asymptotic deviation bounds from bounds on moments of recurrence times]
• Stefano Luzzatto
  • "Mixing and decay of correlations in non-uniformly expanding maps: a survey of recent results," math.DS/0301319
  • "Stochastic-like behaviour in nonuniformly expanding maps", math.DS/0409085
• Stefano Luzzatto, Ian Melbourne and Frederic Paccaut, "The Lorenz Attractor is Mixing", Communications in Mathematical Physics 260 (2005): 393--401
• Vincent Lynch, "Decay of correlations for non-Holder observables", math.DS/0401432
• Michael C. Mackey and Marta Tyran-Kaminska
  • "Deterministic Brownian Motion: The Effects of Perturbing a Dynamical System by a Chaotic Semi-Dynamical System", cond-mat/0408330
  • "Effects of Noise on Entropy Evolution", cond-mat/0501092
  • "Central Limit Theorems for Non-Invertible Measure Preserving Maps", math.PR/0608637 ["a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic, using the Perron-Frobenius operator"]
• P. T. Maker, "The ergodic theorem for a sequence of functions", Duke Mathematical Journal 6 (1940): 27--30
• Katalin Marton and Paul C. Shields, "How many future measures can there be?", Ergodic Theory and Dynamical Systems 22 (2002): 257--280
• Jonathan C. Mattingly, "On Recent Progress for the Stochastic Navier Stokes Equations", math.PR/0409194 ["We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations."]
• Ian Melbourne and Matthew Nicol
  • "Almost Sure Invariance Principle for Nonuniformly Hyperbolic Systems", Communications in Mathematical Physics 260 (2005): 131--146 = math.DS/0503693 ["We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. ... Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences."]
  • "A Vector-Valued Almost Sure Invariance Principle for Hyperbolic Dynamical Systems", math.DS/0606535
• D. S. Ornstein and B. Weiss, "Statistical Properties of Chaotic Systems," Bulletin of the American Mathematical Society 24 (1991): 11--116
• Karl E. Petersen, Ergodic Theory
• Charles Pugh and Michael Shub, with an appendix by Alexander Starkov, "Stable Ergodicity", Bulletin of the American Mathematical Society (new series) 41 (2003): 1--41 [Link]
• Maxim Raginsky, "Joint universal lossy coding and identification of stationary mixing sources with general alphabets", arxiv:0901.1904
• M. Rosenblatt, "Central limit theorem for stationary processes", Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, pp. 551--561
• Gennady Samorodnitsky, "Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes", Annals of Probability 32 (2004): 1438--1468 = math.PR/0410149
• Alexander Schoenhuth, "The ergodic decomposition of asymptotically mean stationary random sources", arxiv:0804.2487
• C. E. Silva, Invitation to Ergodic Theory
• Ya. G. Sinai
  • Introduction to Ergodic Theory
  • Topics in Ergodic Theory
• Charles Stein, "A bound for the error in the normal approximation to the distribution of a sum of dependent random variables", Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, pp. 583--602
• Andre Toom
  • "Law of Large Numbers for Non-Local Functions of Probabilistic Cellular Automata", Journal of Statistical Physics 133 (2008): 883--897
  • "Every Continuous Operator Has an Invariant Measure", Journal of Statistical Physics 129 (2007): 555--566
• Cristina Tone, "Central limit theorems for Hilbert-space valued random fields satisfying a strong mixing condition", arxiv:1012.1842
• Marta Tyran-Kaminska, "Convergence to Lévy stable processes under strong mixing conditions", arxiv:0907.1185
• Marcelo Viana and Kreley Oliveira, Foundations of Ergodic Theory
• R. Vilela Mendes, "Beyond Lyapunov", arxiv:1008.2664 ["Ergodic parameters like the Lyapunov and the conditional exponents are global functions of the invariant measure, but the invariant measure itself contains more information. A more complete characterization of the dynamics by new families of ergodic parameters is discussed, as well as their relation to the dynamical R\'{e}nyi entropies and measures of self-organization"]
• Vladimir V'yugin, "On Instability of the Ergodic Limit Theorems with Respect to Small Violations of Algorithmic Randomness", arxiv:1105.4274
• Peter Walters, An Introduction to Ergodic Theory
• Wei Biao Wu, Xiaofeng Shao, "Invariance principles for fractionally integrated nonlinear processes", math.PR/0608223
• Wei Biao Wu and Michael Woodroofe, "Martingale Approximations for Sums of Stationary Processes", Annals of Probability 32 (2004): 1674--1690 = math.PR/0410160
• Ivan Werner
  • "Ergodic theorem for contractive Markov systems", Nonlinearity 17 (2303--2313) [PS preprint]
  • "Equilibrium states and invariant measures for random dynamical systems", arxiv:1203.6432
• Guangyu Yang, Yu Miao, "An invariance principle for the law of the iterated logarithm for additive functionals of Markov chains", math.PR/0609593
• Radu Zaharopol, Invariant Probabilities of Markov-Feller Operators and Their Supports
• Steve Zelditch, "Quantum ergodicity and mixing", quant-ph/0503026 ["an expository article for the Encyclopedia of Mathematical Physics"]

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Updated 29 October 2007; thanks to "tushar" for pointing out an embarrassing think-o in the first paragraph.

Previous versions: 2007-10-29 9:40; 2005-11-15 16:18; first version written c. 1997 (?)

~A.