Measures of Maximal Entropy for Shifts of Finite Type

Measures of maximal entropy for shifts of finite type Ben Sherman July 7, 2013 Contents Preface 2 1 Introduction 9 2 Dynamical systems 10 3 Topological entropy 13 4 Measure-preserving dynamical systems 20 5 Measure-theoretic entropy 22 6 Parry measure 27 7 Conclusion 30 Appendix: Deriving the Parry measure 30 1 Preface Before I get to the technical material, I thought I might provide a little non-technical background as to why randomness is so interesting, and why unpredictable dynamical systems (i.e., dynamical systems with positive entropy) provide an interesting explanation of randomness. What is randomness? \How dare we speak of the laws of chance? Is not chance the antithesis of all law?" |Joseph Bertrand, Calcul des probabilités, 1889 Throughout history, phenomena which could not be understood in a mundane sense have been ascribed to fate. After all, if something cannot be naturally understood, what else but a supernatural power can explain it? Concepts of chance and randomness thus have been closely associated with actions of deities. After all, for something to be random, it must, by definition, defy all explanation. Many ancient cultures were fascinated with games of chance, such as throwing dice or flipping coins, and interpreted their outcomes as prescriptions of fate. To play a game of chance was to probe the supernatural world. In ancient Rome, the goddess Fortuna was the deity who determined fortune and fate; the two concepts were naturally inseparable. Randomness, in many ways, is simply a lack of understanding. Democritus of ancient Greece realized the subjectivity of randomness with an explanatory story (Wikipedia, 2012). Suppose two men arrange to send their servants to fetch water at the same time. When the servants meet while fetching water, they deem their encounter random, and ascribe it to actions of the Gods, while it is known by the men who sent them that their meeting was arranged by mortals. As Bertrand exclaims, it seems paradoxical to formulate laws that explain the unex- plainable. Perhaps this is why probability theory was so slow to develop. If each random event is determined by fate, it would be unreasonable to expect collections of random 2 events to follow definite, mundane patterns. It was not until 1565, when Gerolamo Car- dano published Liber de Lude Aleae, a gambler's manual that discusses the odds of winning games of chance, that it was realized that a collection of many outcomes of random events follows strong patterns. Cardano noted that as one observes more outcomes of games of chance, the frequencies of the outcomes come closer to certain numbers. Cardano called these special numbers probabilities, and the theory of probability was born. While the development of probability theory answered many questions about the aggre- gated outcomes of random events, it still leaves undecided how the outcome of an individual random event is determined. As a consequence, it also cannot provide an explanation for why the frequencies of outcomes of random events approach the probabilities of those events. Chance remains \the antithesis of all law." There are two directions we can turn to for answers. We can ascribe it to fate, as has been done since the dawn of history. Or, we could turn to the laws of physics. Newton's laws \God doesn't play dice with the world." |Albert Einstein, conversation with William Hermanns, 1943 The world is unpredictable. One could say that we have good fortune that this is true, as a predictable world would certainly not be very exciting. Given our current state, we'd know exactly what would happen in the future. It would be a world without choice and devoid of agency: simply a long, boring march through time. In fact, it would be much worse; in a predictable world, our infinite knowledge of the world's state in the past and future would preclude us from experiencing time, as we would not accumulate memory as time passed by. There would be no way to assert which direction of time would be \forward," and time would reduce to something akin to another spatial dimension. But this putative dull world is not so far-fetched, and in fact it is quite peculiar that our world is not that way. When physicist Isaac Newton devised what are now known 3 as the classical laws of physics (around the year 1700), he essentially claimed that our n world is that bleak predictable one described earlier. Let x 2 R be a vector of spatial coordinates of particles in an isolated system (such as the universe). For example, we could 6 have x 2 R for two particles in 3-dimensional space, with x1 and x4 the x-coordinates of particles 1 and 2, respectively, et. cetera. Let mi be the mass of the particle whose coordinate is described by xi. Newton claimed then that there was some time-independent energy potential U(x) that was a function only of the positions of the particles, such that for all i, 2 @U(x) d xi − = mi 2 : @xi dt Thus, Newton claimed that the trajectory of any system of particles was determined by a system of n ordinary differential equations which are described above. Suppose for a given time t0, we know the positions x0 and velocities x_ 0 of all coordinates. Then existence and uniqueness theorems for ordinary differential equations assure that there is a single unique solution x(t), the trajectory of all the particles through time. Therefore, Newton's laws uniquely determine the trajectory of a system. This means that Newton's laws are deterministic, and leave no room for anything such as randomness. We also observe that Newton's laws are time-symmetric; suppose we have a trajectory x(t) that satisfies Newton's laws. Then one can check that the time-reversed trajectory x~(t) defined by x~(t) = x(−t) also satisfies Newton's laws, and thus is an equally plausible trajectory. Newton's laws have received some adjustments in the 300 years since he first formulated them, but the two principles of determinism and time-symmetry still hold, with some revision. Now, in order to produce a plausible time-reversed trajectory, we also need to mirror charge and parity. But every trajectory still has a plausible time-reversed trajectory. The quantum physical revision states that unobservable information about particles still evolves deterministically, but leaves open to interpretation how these unobservables relate to measured observables of a system. 4 But both classical and quantum mechanics seem to imply that the world evolves deterministically with time. But we are not consigned to the dull world that Newton's laws imply. We don't know the weather two weeks from now, or the winner of tonight's bas- ketball game. We can't even predict the start of an earthquake or volcanic eruption the second before one occurs! What gives? There is one physical \law" that is time-asymmetric and thus discriminates between directions of time (Baranger, 2000). It is the Second Law of Thermodynamics, which states that the entropy of the world does not decrease (and sometimes increases) as time moves \forward." This actually provides the only physical definition for what the \forward" direction of time even means! But what is this nebulous quantity entropy? In the study of thermodynamics, it was empirically discovered that energy, in the form of heat, tends to flow from objects of high temperature to those of low temperature as time advances. Physicist Rudolf Clausius coined the term entropy in 1868, drawing from the Greek word entropia, meaning \a turning toward," for a measure he defined that related to the lack of potential of low-temperature objects to transfer heat energy to high-temperature ones. But alas, temperature was in turn defined in terms of capability to transfer heat, and we are left with little insight. And none of this could be derived from Newton's laws. Physicist Ludwig Boltzmann was the first to rigorously define entropy. Boltzmann is known as the founder of statistical mechanics, a field that uses statistics to reconcile the behavior of microscopic particles with the properties of the bulk material they compose. His tombstone famously bears the equation that encapsulates this founding idea, S = k log W; where S is the entropy of a system, k is a constant (the now-eponymous Boltzmann constant), and W is the number of possible indistinguishable microstates that could equivalently describe the system's state. Therefore, the Second Law equivalently states that the value of W for the universe doesn't decrease as time moves forward. Boltzmann was also 5 able to define temperature of a bulk material in terms of the microscopic states of its con- stituent particles. Boltzmann's entropy managed to suitably explain irreversible processes like heat transfer from hot to cold objects, mixing of two different substances, and phase transitions of substances at given temperatures. Boltzmann's definition is simultaneously problematic and intriguing. First, it is problematic, because calculating the entropy of a system depends very strongly on how a system's state (specifically \macrostate") is to be described. The more accurately we describe the system, the fewer indistinguishable microstates there would be that would count as describing the system, and thus the lower the entropy. So the thermodynamic fact that that Clausius concluded reduces to something that seems surprisingly subjective. Perhaps, as time goes by, physicists simply get lazier and less rigorously describe the macroscopic states of their systems! But it is also offers a very intriguing interpretation of the Second Law: as time goes by, we cannot ever become more capable of describing the world, and sometimes we become less capable.

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