Ergodic Theory

Neste texto, escrito para fins pessoais, detalhamos a demonstração de Artur Ávila e Jairo Bochi [AB] para o Teorema Ergódico Subaditivo devido à Kingman (1968), como caso particular obtemos o teorema ergódico de Birkhoff (1931) em tempo discreto. Além disso, são enunciados e provados os mesmos teoremas para fluxo.

The concept of ergodicity in economics seems to have the qualities of a shibboleth—a word or saying used by adherents of a party, sect, or belief, and usually regarded by others as empty of real meaning. It is in use by both neoclassical economics—after Samuelson (1965, p. 43), who used the term in his paper on what later became a foundation of the efficient market hypothesis— and post Keynesian economics—after Davidson, who picked up the term in order to highlight methodological differences. Considering the origin of the concept in statistical physics and its use in the topology of dynamical systems, which most economists are not conversant with, the importance ascribed to ergodicity in economic debate seems mystifying. We decon-struct the meaning of the term in the major contributions of Samuelson and Davidson. We suggest an alternative to (non) ergodicity to discuss the nature of randomness in the real world. While neoclassical theory assumes stochastic randomness, post Keynesians assume nonstochastic randomness, a term developed by the mathematician Kolmogorov (1986, p. 467). We argue that even in an ergodic world there is a problem with the idea that stochastic randomness can be dealt with by the financial system.

Stories told through different media forms feel very distinctive from each other, to such an extent that there are stories which can only be told through one media form -- at least, if preserving the distinctive affective quality of the experience is a priority. Is this due to something innate to the story which makes it hard to translate outside of its context, or is it the context itself that sets the experience apart? Phenomenology provides a way of understanding how the media-specific structures of textual storytelling can shape the experience of negotiating that text, through altering the affective processes associated with its navigation. This project argues that it is possible to distinguish amongst storytelling in multiple media forms by analysing how differences in textual structure (and the processes required to engage with them) shape the phenomenological experience of those texts. I apply an analytical framework of affective phenomenology to case-study forms of textual storytelling, including videogames, hypertext fiction, webcomics, and Alternate Reality Games. All of these case studies share a new media context, and so their outward similarities will highlight the differences in the experiences they present. I argue that hypertext fictions provide environments for readers to engage with, either as explorers negotiating unfamiliar territory, or detectives seeking connections between disparate material. The webcomic is distinguished from other forms of mediated storytelling by the amount of time spent engaging with characters within the text, which leads to a perception of intimacy as part of the experience. Videogames are set apart by the sense of responsibility felt by the player for events and their consequences within the Heideggerian world-of-concern established with the text and its characters. Alternate Reality Games are texts which function at the level of the community rather than the individual, are experienced as phenomenologically real, and are further distinguished by their textual boundaries functioning at the level of affective investment rather than the specific processes involved in negotiating the text. I argue that the definition of media texts should include how we engage with their textual structures, rather than focus purely on the textual structures themselves. Affective phenomenology and the process of analytical juxtaposition presented in this project provide the beginnings of a map for negotiating this new conceptual territory, and will become particularly relevant as texts and textual forms migrate across platforms.

"Choose policies not for what they intend to do, but for how people react to them."

To be able to design policies, technologies and organizational structures that protect us from risk rather than exposing us to it after second-order effects are taken into account, we need to understand why people and organization take risks and how they become more or less vulnerable to them. People and organization are complex entities, and damage has complex effects upon them. Some forms of damage kill them and other ones make them stronger. Some form of protection protect them, whereas others make them take more risk, eventually resulting in tragedies.

Classical behavioral economics often uses one or more assumptions not representative of the real world: static monolithic players playing single gambles single times with known static odds. This paper discards these assumptions and instead considers complex players made of multiple levels of components (e.g., muscle fibers for humans or citizens for cities) with non-homogenous and anisotropic internal structures, exposed to multiple risks (social, financial, reproductive, etc), engaged in multiple repeated interactions with unknown dynamic odds and who undergo plastic changes in both their physique and in their behavior when damaged or rewarded.

This paper adopts a bottom-up approach to build a coherent and comprehensive framework to estimate the long-term effect of policies, technologies and organizational structures on the risk-taking behavior of people and organization in the real world and on their ultimate survival. In particular, it focuses on how damage can both make them stronger or weaker, more or less prone to take risks, and on what it depends.

ABSTRACT: The free-energy principle states that all systems that minimize their free energy resist a tendency to physical disintegration. Originally proposed to account for perception , learning, and action, the free-energy principle has been applied to the evolution, development, morphology, anatomy and function of the brain, and has been called a postulate, an unfalsifiable principle, a natural law, and an imperative. While it might afford a theoretical foundation for understanding the relationship between environment , life, and mind, its epistemic status is unclear. Also unclear is how the free-energy principle relates to prominent theoretical approaches to life science phenomena, such as organicism and mechanism. This paper clarifies both issues, and identifies limits and prospects for the free-energy principle as a first principle in the life sciences.

A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal R$ on a standard Borel space is amenable if and only if the full group $[{\mathcal R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a nontrivial space in the sense of Gromov occuring in the automorphism groups of injective factors of type $III$.

We prove Sarnak's M\"{o}bius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $\Gamma\subset G$ a lattice, and $u\in G$ an Ad-unipotent element, then for every $x\in\X$ and every continuous, bounded function $f$ on $\X$, the sequence $f(xu^{n})$ cannot correlate with the M\"{o}bius function on average.

We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their orbits coincide. Secondly, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a $u$-Gibbs state and an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strong-unstable direction. This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows.

Unimodal maps have been broadly used as a base of new encryption strategies. Recently, a stream cipher has been proposed in the literature, whose keystream is basically a symbolic sequence of the (one-parameter) logistic map or of the tent map. In the present work a thorough analysis of the keystream is made which reveals the existence of some serious security problems

The digital medium, specially in the case of Netflix, offers a new set of tools ready to revolutionize the world of serial fiction. In this new context, the audience acquires total control over the consumption habits of TV shows: when, where and how many. After being canceled for seven years, Arrested Development is brought back to life within this digital ecosystem. Aware of the possibilities of the medium, this sitcom displays a unique serial structure in which we, the audience, need to interact with the images in order to reconstruct the story: the active viewer is born within a new type of seriality we will refer to as ergodic.

Constructive and computationally tractable techniques are developed for the classification of complex dynamical systems. These techniques subsume and extend standard dynamical characterizations based on Renyi dimensions and entropy spectra, invariant measures, algorithmic complexities and other dynamical and statistical measures. A natural way of quantifying the complexity of spatio-temporal data sets, and hence the combination of underlying physical system and measuring apparatus, is presented. These methods uniquely associate a data set with a formal language, given a specified model basis. The position of this language in a hierarchy of formal languages provides a measure of the complexity of the data set. This in turn provides a quantitative measure of the experimental or computational resources necessary for the complete characterization of a given system. Examples of data sets produced by various systems and models are analyzed using these techniques. It is shown that these techniques are particularly effective in detecting structure and elucidating underlying mechanisms for complex physical phenomena.

This project examined the often competing perspectives of ludology and narratology in the field of video game studies, and attempts to incorporate both into a model of game narrative engagement based on the laws of thermodynamics. Further, this project sought to demonstrate that the same model can be used to examine the study of traditional literary texts.

The Assassin's Creed video game series served as the video game text under examination, and the novel House of Leaves as the literary.

This paper analyzes the use of the logistic map for cryptographic applications. The most important characteristics of the logistic map are shown in order to prove the inconvenience of considering this map in the design of new chaotic cryptosystems.

Various results are presented here: First, a simple but formal measure-theoretic construction of the derivative is given, making it clear that it has a very concrete existence as a Lebesgue-Stieltjes measure, and thus is safe to manipulate in various familiar ways. Next, an exact result is given, expressing the measure as an infinite product of piece-wise continuous functions, with each piece being a Mobius transform of the form (ax+b)/(cx+d). This construction is then shown to be the Haar measure of a certain transfer operator. A general proof is given that any transfer operator can be understood to be nothing more nor less than a push-forward on a Banach space; such push-forwards induce an invariant measure, the Haar measure, of which the Minkowski measure can serve as a prototypical example. Some minor notes pertaining to it's relation to the Gauss-Kuzmin-Wirsing operator are made.

Since 1990s chaotic dynamical systems have been widely used to design new strategies to encrypt information. Indeed, the dependency to initial conditions and control parameters, along with the ergodicity of their temporal evolution allow the establishment of chaos as the base of new cryptosystems, i.e., of new schemes of confusion and diffusion of information. However, an optimum design in the context of chaos-based cryptography demands a thorough knowledge not only of the foundations of cryptography, but also of the dynamics and inner structure of chaos. Therefore, any proposal to use chaos in the context of cryptography must respect a series of design rules, in order to avoid the reconstruction of the dynamics of the underlying chaotic system, and to determine an optimum use of the virtues of the chaotic dynamics. Although it is possible to use chaos to design analog cryptosystems based on synchronization techniques, this Thesis is focused on the application of chaotic maps, i.e., chaotic dynamical systems defined in discrete time to cryptography. In this sense, a set of mathematical tools are defined to establish the adequacy of a chaotic map as the base of a cryptosystem, and the requirements that an encryption architecture must satisfy to avoid the dynamical reconstruction of the underlying chaotic map. More precisely, this Thesis provides an extension and systematization of the results derived from the cryptanalysis of chaos-based cryptosystems. The above goal comprises three different stages: 1.- Definition of a set of mathematical tools that allow the selection of the adequate configurations of a dynamical system to implement strategies of confusion and diffusion of information. 2.- Study of the most popular chaotic maps in the field of chaos-based cryptography to determine whether these maps can be used to design new cryptosystems without incurring in security problems. 3.- Summary and conclusions of the first two stages. The aim is to define a set of rules or recommendations as a guide for the design of chaos-based cryptosystems. Recalling the first stage, its main purpose is the search of procedures to infer or estimate the initial conditions and/or the control parameters from the orbits of a chaotic map. Different scenarios are considered depending on whether complete orbits are accesible or it is only possible to work with sampled or discretized versions of the orbits. In all scenarios the goal consist in building bijective functions with respect to the initial conditions and/or the control parameters. The requirements to build these bijective functions are clarified, along with the procedures to guide the estimation of the initial conditions and/or the control parameters. In order to test the set of mathematical tools and the estimation methods, the logistic map and its associated topological conjugate maps are thoroughly studied, since these maps are the most widely used in the design of new digital chaotic cryptosystems. Specially relevant is the study of the symbolic dynamics and order patterns of unimodal maps. The study of this family of chaotic maps leads to a series of very useful results to define a set of recommendations for both the evaluation of the security of chaos-based cryptosystems and the design of encryption schemes based on chaos.

In this paper we study the ergodic theory of a class of symbolic dynamical systems $(\O, T, \mu)$ where $T:{\O}\to \O$ the left shift transformation on $\O=\prod_0^\infty\{0,1\}$ and $\mu$ is a $\s$-finite $T$-invariant measure having the property that one can find a real number $d$ so that $\mu(\tau^d)=\infty$ but $\mu(\tau^{d-\epsilon}) 0$, where $\tau$ is the first passage time function in the reference state 1. In particular we shall consider invariant measures $\mu$ arising from a potential $V$ which is uniformly continuous but not of summable variation. If $d&gt;0$ then $\mu$ can be normalized to give the unique non-atomic equilibrium probability measure of $V$ for which we compute the (asymptotically) exact mixing rate, of order $n^{-d}$. We also establish the weak-Bernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead $d\leq 0$ then $\mu$ is an infinite measure with scaling rate of order $n^d$. Moreove...

We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice. For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Gamma, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Gamma, and the domains where the averages are supported. In particular, we establish the general principle that under these conditio...

We provide a self-contained, accessible introduction to Ratner&#39;s Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early 1980s by Dani and Smillie and later reappeared as an illustrative special case of Ratner&#39;s work on the equidistribution of unipotent flows in homogeneous spaces. We

Only for ergodic processes will inferences based on group-level data generalize to individual experience or behavior. Because human social and psychological processes typically have an individually variable and time-varying nature, they are unlikely to be ergodic. In this paper, six studies with a repeated-measure design were used for symmetric comparisons of interindividual and intraindividual variation. Our results delineate the potential scope and impact of nonergodic data in human subjects research. Analyses across six samples (with 87–94 participants and an equal number of assessments per participant) showed some degree of agreement in central tendency estimates (mean) between groups and individuals across constructs and data collection paradigms. However, the variance around the expected value was two to four times larger within individuals than within groups. This suggests that literatures in social and medical sciences may overestimate the accuracy of aggregated statistical estimates. This observation could have serious consequences for how we understand the consistency between group and individual correlations, and the gen-eralizability of conclusions between domains. Researchers should explicitly test for equivalence of processes at the individual and group level across the social and medical sciences. research methodology | replicability | idiographic science | generalizability | ecological fallacy I nferences made in social and medical research typically result from statistical tests conducted on aggregated data. The implicit assumption is that group-derived estimates can be applied to understanding individual phenomenology, physiology, and behavior. However, statistical findings at the interindividual (group) level only generalize to the intraindividual (person) level if the processes in question are ergodic (1). Ergodic processes are equivalent for groups and individuals (homogeneity criterion) when their mean and variance remain consistent over time (stationarity criterion) (2). Because psychological and biological phenomena are organized within persons over time, generalizations that rely on group estimates are nonergodic if there are individual exceptions. Group estimates can be derived from a cross-sectional measurement of individuals at one point in time, but intraindividual analyses require data collected over time—as the sample size becomes the number of repeated observations. Just as a random sample is required to be representative of a population to support generalizable claims about that population, the data sampled within an individual in time must be representative of that individual generally (i.e., stationary). When group generalizations obscure genuine individual differences, we may fail to describe natural processes and their natural kinds (3). If scientific consilience (4) and completeness (5) are our goals, this scenario should be avoided. In this paper, we contend that (i) nonergodicity—specifically, the lack of generalizability from group to individual statistical estimates—is a threat to human subjects research, because we do not know the full scope of the problem and are not adequately studying it; and (ii) that scientists need to demonstrate the consistency between individual and group variability before generalizing results across levels of analysis. We will refer to this latter condition as the " group-to-individual generalizability " of a given statistical estimate. However, whether couched in prosaic terms, or within formal mathematical theorems, researchers have not systematically examined such generalizability in extant lit-eratures, despite a number of calls to do so throughout the years (cf. refs. 6–11). Hitherto, the highest-impact publications in medical and social sciences have been largely based on data aggregated across large samples, with best-practice guidelines almost exclusively based on statistical inferences from group designs. The worst-case scenario—a global, uniform absence of group-to-individual generalizability due to nonergodicity in the social and medical sciences—would undermine the validity of our scientific canon in these domains. However, even moderate incongruities between group and individual estimates could result in imprecise or potentially invalid conclusions. We argue that this possibility should be formally tested, wherever possible, to be ruled out. Ergodicity, the Ecological Fallacy, and Simpson's Paradox The ergodic theorem is a general and formal mathematical expression that deals with the generalizability of statistical phenomena across levels and units of analysis. [While a more thorough explication of the ergodic theorem is outside of the scope of the present paper, readers are referred to Molenaar (1) for a comprehensive mathematical treatment of ergodicity in human subjects research.] Ergodic theory postulates that the Significance The current study quantified the degree to which group data are able to describe individual participants. We utilized intensive repeated-measures data—data that have been collected many times, across many individuals—to compare the distributions of bivariate correlations calculated within subjects vs. those calculated between subjects. Because the vast majority of social and medical science research aggregates across subjects, we aimed to assess how closely such aggregations reflect their constituent individuals. We provide evidence that conclusions drawn from aggregated data may be worryingly imprecise. Specifically, the variance in individuals is up to four times larger than in groups. These data call for a focus on idiography and open science that may substantially alter best-practice guidelines in the medical and behavioral sciences.

This survey is focused on the results related to topologies on the groups of transformations in ergodic theory, Borel, and Cantor dynamics. Various topological properties (density, connectedness, genericity) of these groups and their subsets (subgroups) are studied.

We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit integrals over moduli spaces of algebraic curves with additional structures. Moreover, these integrals can be interpreted as correlators in a topological string theory. Also a new analogy arose between ergodic theory and complex algebraic geometry.

We formulate and prove a {\it Local Stable Manifold Theorem\/} for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and It\^o-type equations are treated. Starting with the existence of a stochastic flow for a sde, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich sde's, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde. The proof of the stable manifold theorem is based on Ruelle-Oseledec multiplicative ergodic theory.

We study deterministic and stochastic perturbations of incompressible flows on a two-dimensional torus. Even in the case of purely deterministic perturbations, the long-time behavior of such flows can be stochastic. The stochasticity is caused by instabilities near the saddle points as well as by the ergodic component of the locally Hamiltonian system on the torus.