4629:
2507:
4277:
5877:. A number of classification theorems have been obtained; but quite interestingly, a number of anti-classification theorems have been found as well. The anti-classification theorems state that there are more than a countable number of isomorphism classes, and that a countable amount of information is not sufficient to classify isomorphisms.
4624:{\displaystyle {\begin{aligned}\bigvee _{n=0}^{N}T^{-n}Q&=\{Q_{i_{0}}\cap T^{-1}Q_{i_{1}}\cap \cdots \cap T^{-N}Q_{i_{N}}\\&{}\qquad {\mbox{ where }}i_{\ell }=1,\ldots ,k,\ \ell =0,\ldots ,N,\ \\&{}\qquad \qquad \mu \left(Q_{i_{0}}\cap T^{-1}Q_{i_{1}}\cap \cdots \cap T^{-N}Q_{i_{N}}\right)>0\}\\\end{aligned}}}
2497:
This system does exhibit one key idea from the classification of measure-preserving dynamical systems: two ensembles, having different temperatures, are inequivalent. The entropy for a given canonical ensemble depends on its temperature; as physical systems, it is "obvious" that when the temperatures
4975:
2414:
of these. This measure is understood to apply to the ensemble. So, for example, one of the possible boxes in the ensemble has all of the atoms on one side of the box. One can compute the likelihood of this, in the
MaxwellâBoltzmann measure. It will be enormously tiny, of order
4262:
755:
4769:
5120:
470:
1884:
is all that is left, after all of the transient modes have decayed away. The transient modes are precisely those eigenvectors of the transfer operator that have eigenvalue less than one; the invariant measure
4848:
2110:
2112:
The "ensemble" is the collection of all such points, that is, the collection of all such possible boxes (of which there are an uncountably-infinite number). This ensemble of all-possible-boxes is the space
2281:
2493:
or some other interaction suitable for a liquid or a plasma; in such cases, the invariant measure is no longer the
MaxwellâBoltzmann distribution. The art of physics is finding reasonable approximations.
4055:
2020:
2754:
5330:
4282:
2464:
5851:
2489:
is difficult, and, even if written down, it is hard to perform practical computations with it. Difficulties are compounded if there are interactions between the particles themselves, like a
253:
5577:
1905:
is the one mode that does not decay away. The rate of decay of the transient modes are given by (the logarithm of) their eigenvalues; the eigenvalue one corresponds to infinite half-life.
3181:
1303:
1134:
5757:
5495:
5034:
4820:
3944:
3719:
3386:
3334:
3282:
3018:
2970:
1882:
1798:
127:
5639:
2823:
5689:
2372:
1643:
1600:
of the transfer operator (recall, the FP eigenvector is the largest eigenvector of a matrix; in this case it is the eigenvector which has the eigenvalue one: the invariant measure.)
1450:
1046:
542:
1351:
5220:
3849:
327:
3124:
4111:
3627:
3540:
2554:
1950:
1703:
595:
2880:
635:
361:
6028:
3053:
1500:
5926:
5875:
5448:
4674:
2610:
the flow of a
Hamiltonian vector field on the tangent bundle of a closed connected smooth manifold is measure-preserving (using the measure induced on the Borel sets by the
1399:
1375:
1161:
1073:
179:
3449:
5382:
3410:
3087:
1542:
1745:
that are measure-like. Measure-like, in that they preserve the Borel properties, but are no longer invariant; they are in general dissipative and so give insights into
983:
944:
292:
1245:
5783:
3589:
3502:
3232:
6097:
5522:
5240:
3563:
3476:
3205:
2158:
1903:
1743:
1590:
389:
5809:
6117:
6068:
6048:
5898:
5709:
5422:
5402:
5192:
2487:
2412:
2392:
2301:
2131:
2040:
1970:
1834:
1723:
1663:
1570:
1205:
1185:
203:
153:
903:
863:
831:
791:
630:
5986:
7374:
4682:
7452:
2559:
Unlike the informal example above, the examples below are sufficiently well-defined and tractable that explicit, formal computations can be performed.
7469:
5042:
5649:
One of the primary activities in the study of measure-preserving systems is their classification according to their properties. That is, let
6237:
394:
6314:
5936:, it can be shown that there are uncountably many non-Kakutani equivalent ergodic measure-preserving transformations of each entropy type.
5245:
4970:{\displaystyle h_{\mu }(T,{\mathcal {Q}})=\lim _{N\rightarrow \infty }{\frac {1}{N}}H\left(\bigvee _{n=0}^{N}T^{-n}{\mathcal {Q}}\right).}
6459:
6186:
2049:
2171:
6777:
6636:
3960:
6223:
5242:
is absolutely continuous with respect to the
Lebesgue measure, then we have the Rokhlin formula (section 4.3 and section 12.3 ):
5933:
5133:
in 1959 shows that the supremum is actually obtained on partitions that are generators. Thus, for example, the entropy of the
1975:
7292:
1917:
from physics provides an informal example. Consider, for example, a fluid, gas or plasma in a box of width, length and height
7123:
6540:
6247:
2702:
1972:
atoms. A single atom in that box might be anywhere, having arbitrary velocity; it would be represented by a single point in
6663:
2615:
2600:
50:. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from
7284:
2418:
7070:
2161:
7464:
5817:
6596:
6568:
6406:
6168:
1548:. Almost all properties and behaviors of dynamical systems are defined in terms of the pushforward. For example, the
210:
7540:
7421:
7411:
5527:
2586:
985:. For the paint thickness to remain unchanged (measure-preserving), the mass of incoming paint should be the same:
3129:
7221:
7130:
6894:
1250:
1085:
6591:, Andreas Greven, Gerhard Keller, and Gerald Warnecke, eds. Princeton University Press, Princeton, NJ (2003).
5714:
4991:
4777:
3901:
3676:
3343:
3291:
3237:
2975:
2927:
1839:
1755:
84:
6750:
6135:
5582:
2769:
43:
7459:
7406:
7300:
7206:
6129:
5652:
2626:
2306:
1606:
1404:
988:
484:
6263:
Foreman, Matthew; Weiss, Benjamin (2019). "From
Odometers to Circular Systems: A Global Structure Theorem".
1308:
7325:
7305:
7269:
7193:
6913:
6629:
5197:
4257:{\displaystyle Q\vee R=\{Q_{i}\cap R_{j}\mid i=1,\ldots ,k,\ j=1,\ldots ,m,\ \mu (Q_{i}\cap R_{j})>0\}.}
1597:
5453:
3801:
750:{\displaystyle Tx=2x\mod 1={\begin{cases}2x{\text{ if }}x<1/2\\2x-1{\text{ if }}x>1/2\\\end{cases}}}
297:
7545:
7447:
7226:
7188:
7140:
3099:
6206:
3594:
3507:
2521:
1920:
1672:
905:
half as well. The two layers of thin paint, layered together, recreates the exact same paint thickness.
547:
7530:
7352:
7320:
7310:
7231:
7198:
6829:
6738:
2834:
334:
7369:
7274:
7050:
6978:
5994:
4634:
which plays crucial role in the construction of the measure-theoretic entropy of a dynamical system.
3026:
2642:
is not a single transformation that is iterated to give the dynamics of the system, but instead is a
2498:
differ, so do the systems. This holds in general: systems with different entropy are not isomorphic.
1801:
1457:
59:
7359:
5932:. There are a variety of other anti-classification results. For example, replacing isomorphism with
5907:
5856:
5427:
5404:
has full measure or zero measure. Piecewise expanding and Markov means that there is a partition of
4655:
1380:
1356:
1142:
1054:
671:
160:
7442:
6888:
6819:
6559:
Michael S. Keane, "Ergodic theory and subshifts of finite type", (1991), appearing as
Chapter 2 in
2572:
2469:
The only reason that this is an "informal example" is because writing down the transition function
6755:
3422:
7550:
7211:
6969:
6929:
6622:
6487:
5342:
2604:
2593:
1914:
6138: â Certain dynamical systems will eventually return to (or approximate) their initial state
3394:
3069:
2638:
The definition of a measure-preserving dynamical system can be generalized to the case in which
1509:
7494:
7394:
7216:
6938:
6784:
6563:, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991).
6478:
6394:
3634:
949:
911:
262:
2629:
establishes the existence of a suitable measure to form a measure-preserving dynamical system.
2611:
1218:
7055:
7008:
7003:
6998:
6840:
6723:
6681:
6516:
Katok, A.; Hasselblatt, B. (1995). "Introduction to the modern theory of dynamical systems".
5762:
3568:
3481:
3211:
7364:
7330:
7238:
6948:
6903:
6745:
6668:
6076:
5943:
Ergodic measure-preserving transformations with a pure point spectrum have been classified.
5812:
5500:
4102:
1805:
5225:
3548:
3461:
3190:
2143:
1888:
1728:
1575:
374:
8:
7347:
7337:
7183:
7147:
6973:
6702:
6659:
5788:
5168:
4649:
4643:
3090:
2647:
2490:
1503:
1208:
481:
One may ask why the measure preserving transformation is defined in terms of the inverse
364:
256:
51:
47:
7025:
7499:
7259:
7244:
6943:
6824:
6802:
6439:
6376:
6323:
6290:
6272:
6102:
6053:
6033:
5883:
5694:
5407:
5387:
5177:
4061:
3740:
2472:
2397:
2377:
2286:
2116:
2025:
1955:
1819:
1746:
1708:
1648:
1555:
1190:
1170:
188:
138:
55:
6099:, then such a generator exists iff the system is isomorphic to the Bernoulli shift on
868:
836:
796:
764:
603:
7416:
7152:
7113:
7108:
7015:
6933:
6718:
6691:
6592:
6564:
6536:
6501:
6482:
6402:
6294:
6243:
6164:
5134:
3855:
3187:
2757:
1666:
1593:
1549:
368:
75:
71:
35:
6380:
6242:. London Mathematical Society Student Texts. Cambridge: Cambridge University Press.
7535:
7433:
7342:
7118:
7103:
7093:
7078:
7045:
7040:
7030:
6908:
6883:
6698:
6535:. New Mathematical Monographs. Cambridge: Cambridge University Press. p. 106.
6496:
6457:
Sinai, Ya. (1962). "A weak isomorphism of transformations with invariant measure".
6431:
6422:
Halmos, P.; von
Neumann, J. (1942). "Operator methods in classical mechanics. II".
6366:
6333:
6282:
5971:
5145:
3658:
2579:
2511:
6580:
4764:{\displaystyle H({\mathcal {Q}})=-\sum _{Q\in {\mathcal {Q}}}\mu (Q)\log \mu (Q).}
7509:
7489:
7264:
7162:
7157:
7135:
6993:
6958:
6878:
6772:
6584:
5950:
5946:
3662:
2165:
7399:
7254:
7249:
7060:
7035:
6988:
6918:
6898:
6858:
6848:
6645:
6576:
1813:
1603:
There are two classification problems of interest. One, discussed below, fixes
39:
20:
2466:
Of all possible boxes in the ensemble, this is a ridiculously small fraction.
7524:
7504:
7167:
7088:
7083:
6983:
6953:
6923:
6873:
6868:
6863:
6853:
6767:
6686:
6572:(Provides expository introduction, with exercises, and extensive references.)
5901:
5149:
1163:
which preserve intersections, unions and complements (so that it is a map of
758:
182:
1804:. One might ask: how did it get that way? Often, the answer is by stirring,
7098:
7020:
6760:
5333:
5156:
is either less than 1/2 or not; and likewise so is the fractional part of 2
5138:
5115:{\displaystyle h_{\mu }(T)=\sup _{\mathcal {Q}}h_{\mu }(T,{\mathcal {Q}}).}
2914:
2826:
2651:
2622:
6797:
1211:). Every such conservative, Borel-preserving map can be specified by some
19:"Area-preserving map" redirects here. For the map projection concept, see
6963:
5141:
5130:
2918:
2564:
27:
6338:
6309:
6286:
6184:
Sinai, Ya. G. (1959). "On the Notion of
Entropy of a Dynamical System".
6807:
6610:(gives a more involved example of measure-preserving dynamical system.)
6443:
5939:
These stand in contrast to the classification theorems. These include:
5167:
is compact and endowed with a topology, or is a metric space, then the
1809:
1800:
often describes a physical system that is in equilibrium, for example,
1212:
6371:
6354:
5332:
This allows calculation of entropy of many interval maps, such as the
465:{\displaystyle \forall A\in {\mathcal {B}}\;\;\mu (T^{-1}(A))=\mu (A)}
6789:
6733:
6728:
6606:
5929:
5880:
The first anti-classification theorem, due to Hjorth, states that if
2137:
1164:
1076:
6435:
6310:"Measure preserving Diffeomorphisms of the Torus are unclassifiable"
3854:
The set of symbolic names with respect to a partition is called the
2506:
6814:
6673:
6328:
6277:
5126:
1545:
5644:
6614:
5968:
Given a dynamical system on a
Lebesgue space of measure 1, where
1552:
is defined in terms of the pushforward of the transformation map
2105:{\displaystyle (w\times l\times h)^{N}\times \mathbb {R} ^{3N}.}
2886:
The earlier, simpler case fits into this framework by defining
2643:
1645:
and asks about the isomorphism classes of a transformation map
2276:{\displaystyle p_{i}(x,y,z,v_{x},v_{y},v_{z})\,d^{3}x\,d^{3}p}
761:. Now, distribute an even layer of paint on the unit interval
5129:
is taken over all finite measurable partitions. A theorem of
1752:
In terms of physics, the measure-preserving dynamical system
4050:{\displaystyle T^{-1}Q=\{T^{-1}Q_{1},\ldots ,T^{-1}Q_{k}\}.}
1353:, but this is not enough to specify all such possible maps
743:
2696:
above. In particular, the transformations obey the rules:
2015:{\displaystyle w\times l\times h\times \mathbb {R} ^{3}.}
6561:
Ergodic Theory, Symbolic
Dynamics and Hyperbolic Spaces
6483:"Bernoulli shifts with the same entropy are isomorphic"
2633:
6355:"On invariants for measure preserving transformations"
5974:
5424:
into finitely many open intervals, such that for some
4430:
2749:{\displaystyle T_{0}=\mathrm {id} _{X}:X\rightarrow X}
1836:
describes this stirring, mixing, etc. then the system
908:
More generally, the paint that would arrive at subset
70:
A measure-preserving dynamical system is defined as a
6105:
6079:
6056:
6036:
5997:
5910:
5886:
5859:
5820:
5791:
5765:
5717:
5697:
5655:
5585:
5530:
5503:
5456:
5430:
5410:
5390:
5345:
5248:
5228:
5200:
5180:
5045:
4994:
4851:
4780:
4685:
4658:
4280:
4114:
3963:
3904:
3804:
3679:
3597:
3571:
3551:
3510:
3484:
3464:
3425:
3397:
3346:
3294:
3240:
3214:
3193:
3132:
3102:
3072:
3029:
2978:
2930:
2837:
2772:
2705:
2654:
upon the given probability space) of transformations
2563:ÎŒ could be the normalized angle measure dΞ/2Ï on the
2524:
2475:
2421:
2400:
2380:
2309:
2289:
2174:
2146:
2119:
2052:
2028:
1978:
1958:
1923:
1891:
1842:
1822:
1758:
1731:
1711:
1675:
1651:
1609:
1578:
1558:
1512:
1460:
1407:
1383:
1359:
1311:
1253:
1221:
1193:
1173:
1145:
1088:
1057:
991:
952:
914:
871:
839:
799:
767:
638:
606:
550:
487:
397:
377:
337:
300:
265:
213:
191:
163:
141:
87:
78:
transformation on it. In more detail, it is a system
34:
is an object of study in the abstract formulation of
6603:
The entropy of strange billiards inside n-simplexes.
6393:
5325:{\displaystyle h_{\mu }(T)=\int \ln |dT/dx|\mu (dx)}
4774:
The measure-theoretic entropy of a dynamical system
3751:measurable pair-wise disjoint sets. Given a point
2459:{\displaystyle {\mathcal {O}}\left(2^{-3N}\right).}
793:, and then map the paint forward. The paint on the
42:in particular. Measure-preserving systems obey the
6111:
6091:
6062:
6042:
6022:
5980:
5920:
5892:
5869:
5845:
5803:
5777:
5751:
5703:
5683:
5633:
5571:
5516:
5497:on each open interval. Markov means that for each
5489:
5442:
5416:
5396:
5376:
5324:
5234:
5214:
5194:is an ergodic, piecewise expanding, and Markov on
5186:
5114:
5028:
4969:
4814:
4763:
4668:
4623:
4256:
4049:
3938:
3843:
3713:
3621:
3583:
3557:
3534:
3496:
3470:
3443:
3404:
3380:
3328:
3276:
3226:
3199:
3175:
3118:
3081:
3047:
3012:
2964:
2874:
2817:
2748:
2548:
2481:
2458:
2406:
2386:
2366:
2295:
2275:
2152:
2125:
2104:
2034:
2014:
1964:
1944:
1897:
1876:
1828:
1792:
1737:
1717:
1697:
1657:
1637:
1584:
1564:
1536:
1494:
1444:
1393:
1369:
1345:
1297:
1239:
1199:
1179:
1155:
1128:
1067:
1040:
977:
938:
897:
857:
825:
785:
749:
624:
600:Consider the typical measure on the unit interval
589:
536:
464:
383:
355:
321:
286:
247:
197:
173:
147:
121:
6421:
6399:Equivalence of measure preserving transformations
3774:can belong to only one of the parts as well. The
2592:with the definition of an appropriate measure, a
1816:or other such processes. If a transformation map
7522:
6518:Encyclopedia of Mathematics and its Applications
6515:
6401:. Mem. American Mathematical Soc. Vol. 37.
5988:is invertible, measure preserving, and ergodic.
5846:{\displaystyle {\mathcal {R}}\subset U\times U.}
5069:
4885:
3668:
3062:if it satisfies the following three properties:
5645:Classification and anti-classification theorems
1377:. That is, conservative, Borel-preserving maps
248:{\displaystyle \mu :{\mathcal {B}}\rightarrow }
3637:of dynamical systems and their homomorphisms.
3416:if, in addition, there exists another mapping
2685:âȘ {0}, or [0, +â)), where each transformation
6630:
6235:
6154:
6152:
5711:be the set of all measure preserving systems
5572:{\displaystyle T(I_{i})\cap I_{i}=\emptyset }
3454:that is also a homomorphism, which satisfies
7375:RieszâMarkovâKakutani representation theorem
6315:Journal of the European Mathematical Society
6307:
6262:
5949:are classified by their metric entropy. See
4637:
4614:
4329:
4248:
4127:
4041:
3983:
3176:{\displaystyle \mu (\varphi ^{-1}B)=\nu (B)}
6530:
6520:. Vol. 54. Cambridge University Press.
3877:
1401:cannot, in general, be written in the form
1298:{\displaystyle {\mathcal {T}}(A)=T^{-1}(A)}
1129:{\displaystyle {\mathcal {T}}:P(X)\to P(X)}
7470:Vitale's random BrunnâMinkowski inequality
6637:
6623:
6308:Foreman, Matthew; Weiss, Benjamin (2022).
6149:
5853:The goal is then to describe the relation
3401:
2882:, whenever all the terms are well-defined.
415:
414:
6500:
6370:
6337:
6327:
6276:
5752:{\displaystyle (X,{\mathcal {B}},\mu ,T)}
5208:
5029:{\displaystyle (X,{\mathcal {B}},T,\mu )}
4815:{\displaystyle (X,{\mathcal {B}},T,\mu )}
3939:{\displaystyle (X,{\mathcal {B}},T,\mu )}
3714:{\displaystyle (X,{\mathcal {B}},T,\mu )}
3381:{\displaystyle (X,{\mathcal {A}},\mu ,T)}
3329:{\displaystyle (Y,{\mathcal {B}},\nu ,S)}
3277:{\displaystyle \varphi (Tx)=S(\varphi x)}
3013:{\displaystyle (Y,{\mathcal {B}},\nu ,S)}
2965:{\displaystyle (X,{\mathcal {A}},\mu ,T)}
2542:
2541:
2394:atoms, the probability is the product of
2259:
2245:
2086:
1999:
1877:{\displaystyle (X,{\mathcal {B}},\mu ,T)}
1793:{\displaystyle (X,{\mathcal {B}},\mu ,T)}
659:
658:
122:{\displaystyle (X,{\mathcal {B}},\mu ,T)}
6477:
5634:{\displaystyle T(I_{i})\cap I_{i}=I_{i}}
2818:{\displaystyle T_{s}\circ T_{t}=T_{t+s}}
2505:
6236:Pollicott, Mark; Yuri, Michiko (1998).
6158:
5684:{\displaystyle (X,{\mathcal {B}},\mu )}
5152:into the intervals . Every real number
2367:{\displaystyle x,y,z,v_{x},v_{y},v_{z}}
1638:{\displaystyle (X,{\mathcal {B}},\mu )}
1445:{\displaystyle {\mathcal {T}}(A)=T(A);}
1041:{\displaystyle \mu (A)=\mu (T^{-1}(A))}
597:. This can be understood intuitively.
537:{\displaystyle \mu (T^{-1}(A))=\mu (A)}
7523:
6352:
6132:on the existence of invariant measures
3858:of the dynamical system. A partition
1346:{\displaystyle {\mathcal {T}}(A)=T(A)}
1139:Consider now the special case of maps
544:instead of the forward transformation
6618:
6456:
6204:
6183:
5215:{\displaystyle X\subset \mathbb {R} }
7483:Applications & related
6605:J. Phys. A 28(17), page 5033, 1995.
6239:Dynamical Systems and Ergodic Theory
6224:The Shannon-McMillan-Breiman Theorem
6207:"Metric Entropy of Dynamical System"
5490:{\displaystyle |T'|\geq 1+\epsilon }
3844:{\displaystyle T^{n}x\in Q_{a_{n}}.}
2634:Generalization to groups and monoids
322:{\displaystyle \mu (\varnothing )=0}
6073:If the entropy is exactly equal to
3119:{\displaystyle B\in {\mathcal {B}}}
2692:satisfies the same requirements as
1908:
1305:. Of course, one could also define
32:measure-preserving dynamical system
13:
6644:
6579:, "Entropy in Dynamical Systems" (
6553:
5913:
5862:
5823:
5729:
5667:
5566:
5524:from those open intervals, either
5101:
5074:
5006:
4954:
4895:
4873:
4792:
4721:
4694:
4661:
4269:refinement of an iterated pullback
3916:
3691:
3622:{\displaystyle y=\varphi (\psi y)}
3535:{\displaystyle x=\psi (\varphi x)}
3358:
3306:
3111:
2990:
2942:
2724:
2721:
2549:{\displaystyle x\mapsto 2x\mod 1.}
2424:
1945:{\displaystyle w\times l\times h,}
1854:
1770:
1698:{\displaystyle (X,{\mathcal {B}})}
1687:
1621:
1410:
1386:
1362:
1314:
1256:
1148:
1091:
1060:
833:half is spread thinly over all of
590:{\displaystyle \mu (T(A))=\mu (A)}
409:
398:
222:
166:
99:
16:Subject of study in ergodic theory
14:
7562:
6397:; Rudolph, D.; Weiss, B. (1982).
6161:An Introduction to Ergodic Theory
5148:. That is, one may partition the
3770:. Similarly, the iterated point
3640:
3060:homomorphism of dynamical systems
2875:{\displaystyle T_{s}^{-1}=T_{-s}}
2616:Liouville's theorem (Hamiltonian)
307:
7412:Lebesgue differentiation theorem
7293:Carathéodory's extension theorem
5959:Krieger finite generator theorem
3782:, with regards to the partition
3414:isomorphism of dynamical systems
2908:
2587:interval exchange transformation
356:{\displaystyle T:X\rightarrow X}
6524:
6509:
6471:
6450:
6415:
6023:{\displaystyle h_{T}\leq \ln k}
4505:
4504:
4428:
4267:With these two constructs, the
3786:, is the sequence of integers {
3048:{\displaystyle \varphi :X\to Y}
2924:Consider two dynamical systems
2537:
1495:{\displaystyle \mu (T^{-1}(A))}
654:
58:systems) as well as systems in
6601:T. SchĂŒrmann and I. Hoffmann,
6587:), appearing as Chapter 16 in
6387:
6346:
6301:
6256:
6229:
6216:
6198:
6177:
5921:{\displaystyle {\mathcal {R}}}
5870:{\displaystyle {\mathcal {R}}}
5746:
5718:
5678:
5656:
5602:
5589:
5547:
5534:
5471:
5458:
5443:{\displaystyle \epsilon >0}
5365:
5359:
5319:
5310:
5303:
5281:
5265:
5259:
5106:
5090:
5062:
5056:
5023:
4995:
4892:
4878:
4862:
4809:
4781:
4755:
4749:
4737:
4731:
4699:
4689:
4669:{\displaystyle {\mathcal {Q}}}
4239:
4213:
3933:
3905:
3708:
3680:
3616:
3607:
3529:
3520:
3435:
3375:
3347:
3323:
3295:
3271:
3262:
3253:
3244:
3170:
3164:
3155:
3136:
3039:
3007:
2979:
2959:
2931:
2740:
2528:
2242:
2185:
2162:MaxwellâBoltzmann distribution
2072:
2053:
1871:
1843:
1787:
1759:
1749:and the route to equilibrium.
1692:
1676:
1632:
1610:
1531:
1528:
1522:
1516:
1489:
1486:
1480:
1464:
1436:
1430:
1421:
1415:
1394:{\displaystyle {\mathcal {T}}}
1370:{\displaystyle {\mathcal {T}}}
1340:
1334:
1325:
1319:
1292:
1286:
1267:
1261:
1231:
1156:{\displaystyle {\mathcal {T}}}
1123:
1117:
1111:
1108:
1102:
1068:{\displaystyle {\mathcal {T}}}
1035:
1032:
1026:
1010:
1001:
995:
972:
966:
933:
921:
892:
872:
852:
840:
820:
800:
780:
768:
619:
607:
584:
578:
569:
566:
560:
554:
531:
525:
516:
513:
507:
491:
459:
453:
444:
441:
435:
419:
347:
310:
304:
275:
269:
242:
230:
227:
174:{\displaystyle {\mathcal {B}}}
132:with the following structure:
116:
88:
1:
6142:
6119:symbols with equal measures.
6050:, then the system has a size-
3669:Symbolic names and generators
2825:, whenever all the terms are
2303:having position and velocity
476:
65:
6533:Entropy in dynamical systems
6531:Downarowicz, Tomasz (2011).
6502:10.1016/0001-8708(70)90029-0
5691:be a measure space, and let
4822:with respect to a partition
3874:has a unique symbolic name.
3673:Consider a dynamical system
3444:{\displaystyle \psi :Y\to X}
2650:, in which case we have the
1598:FrobeniusâPerron eigenvector
1592:can now be understood as an
46:, and are a special case of
7:
7465:PrĂ©kopaâLeindler inequality
6136:Poincaré recurrence theorem
6123:
5377:{\displaystyle T^{-1}(A)=A}
3763:belongs to only one of the
2501:
2283:is the probability of atom
1665:. The other, discussed in
44:Poincaré recurrence theorem
10:
7567:
7407:Lebesgue's density theorem
6460:Doklady Akademii Nauk SSSR
6265:Journal of Modern Dynamics
6187:Doklady Akademii Nauk SSSR
4641:
3665:according to the measure.
3405:{\displaystyle \varphi \;}
3082:{\displaystyle \varphi \ }
1537:{\displaystyle \mu (T(A))}
1207:(because we want it to be
18:
7482:
7460:MinkowskiâSteiner formula
7430:
7390:
7383:
7283:
7275:Projection-valued measure
7176:
7069:
6838:
6711:
6652:
6130:KrylovâBogolyubov theorem
4986:measure-theoretic entropy
4638:Measure-theoretic entropy
3898:} and a dynamical system
2627:KrylovâBogolyubov theorem
2491:van der Waals interaction
1802:thermodynamic equilibrium
978:{\displaystyle T^{-1}(A)}
939:{\displaystyle A\subset }
287:{\displaystyle \mu (X)=1}
60:thermodynamic equilibrium
7443:Isoperimetric inequality
7422:VitaliâHahnâSaks theorem
6751:Carathéodory's criterion
3878:Operations on partitions
3870:if Ό-almost every point
2573:equidistribution theorem
1544:is generically called a
1240:{\displaystyle T:X\to X}
7541:Entropy and information
7448:BrunnâMinkowski theorem
7317:Decomposition theorems
6488:Advances in Mathematics
6159:Walters, Peter (2000).
5785:of two transformations
5778:{\displaystyle S\sim T}
4982:KolmogorovâSinai metric
3882:Given a partition Q = {
2605:random dynamical system
2594:subshift of finite type
2518: : [0,1) â [0,1),
2046:somewhere in the space
1915:microcanonical ensemble
865:, and the paint on the
7495:Descriptive set theory
7395:Disintegration theorem
6830:Universally measurable
6205:Sinai, Ya. G. (2007).
6113:
6093:
6064:
6044:
6024:
5982:
5922:
5894:
5871:
5847:
5805:
5779:
5753:
5705:
5685:
5635:
5573:
5518:
5491:
5444:
5418:
5398:
5378:
5326:
5236:
5216:
5188:
5116:
5030:
4988:of a dynamical system
4971:
4938:
4816:
4765:
4670:
4625:
4305:
4258:
4051:
3940:
3845:
3715:
3633:Hence, one may form a
3623:
3585:
3584:{\displaystyle y\in Y}
3559:
3536:
3498:
3497:{\displaystyle x\in X}
3472:
3445:
3406:
3382:
3330:
3278:
3228:
3227:{\displaystyle x\in X}
3201:
3177:
3120:
3083:
3049:
3014:
2966:
2876:
2819:
2750:
2612:symplectic volume form
2556:
2550:
2483:
2460:
2408:
2388:
2368:
2297:
2277:
2154:
2127:
2106:
2042:atoms would then be a
2036:
2022:A given collection of
2016:
1966:
1946:
1899:
1878:
1830:
1794:
1739:
1725:, and asks about maps
1719:
1699:
1659:
1639:
1586:
1566:
1538:
1496:
1446:
1395:
1371:
1347:
1299:
1241:
1201:
1181:
1157:
1130:
1069:
1042:
979:
946:comes from the subset
940:
899:
859:
827:
787:
751:
626:
591:
538:
466:
385:
357:
323:
288:
249:
199:
175:
149:
123:
7297:Convergence theorems
6756:Cylindrical Ï-algebra
6424:Annals of Mathematics
6114:
6094:
6092:{\displaystyle \ln k}
6065:
6045:
6025:
5983:
5923:
5895:
5872:
5848:
5806:
5780:
5754:
5706:
5686:
5636:
5574:
5519:
5517:{\displaystyle I_{i}}
5492:
5445:
5419:
5399:
5379:
5327:
5237:
5217:
5189:
5171:may also be defined.
5137:is log 2, since
5117:
5031:
4972:
4918:
4842:} is then defined as
4817:
4766:
4671:
4626:
4285:
4259:
4052:
3941:
3846:
3716:
3663:distributed uniformly
3624:
3586:
3560:
3537:
3499:
3473:
3446:
3407:
3383:
3331:
3279:
3229:
3202:
3178:
3121:
3084:
3050:
3015:
2967:
2877:
2820:
2751:
2621:for certain maps and
2551:
2509:
2484:
2461:
2409:
2389:
2369:
2298:
2278:
2155:
2128:
2107:
2037:
2017:
1967:
1947:
1900:
1879:
1831:
1795:
1740:
1720:
1700:
1660:
1640:
1587:
1567:
1539:
1497:
1447:
1396:
1372:
1348:
1300:
1242:
1202:
1182:
1158:
1131:
1070:
1043:
980:
941:
900:
860:
828:
788:
752:
627:
592:
539:
467:
386:
367:transformation which
358:
324:
289:
250:
200:
176:
150:
124:
54:(in particular, most
7365:Minkowski inequality
7239:Cylinder set measure
7124:Infinite-dimensional
6739:equivalence relation
6669:Lebesgue integration
6103:
6077:
6054:
6034:
5995:
5972:
5934:Kakutani equivalence
5908:
5900:is endowed with the
5884:
5857:
5818:
5813:equivalence relation
5789:
5763:
5715:
5695:
5653:
5583:
5528:
5501:
5454:
5428:
5408:
5388:
5343:
5246:
5235:{\displaystyle \mu }
5226:
5198:
5178:
5043:
4992:
4849:
4778:
4683:
4656:
4278:
4112:
3961:
3902:
3868:generating partition
3802:
3677:
3595:
3569:
3558:{\displaystyle \nu }
3549:
3508:
3482:
3471:{\displaystyle \mu }
3462:
3423:
3395:
3344:
3292:
3238:
3212:
3200:{\displaystyle \mu }
3191:
3130:
3100:
3070:
3027:
2976:
2928:
2835:
2770:
2703:
2522:
2473:
2419:
2398:
2378:
2307:
2287:
2172:
2153:{\displaystyle \mu }
2144:
2117:
2050:
2026:
1976:
1956:
1921:
1898:{\displaystyle \mu }
1889:
1840:
1820:
1756:
1738:{\displaystyle \mu }
1729:
1709:
1673:
1649:
1607:
1585:{\displaystyle \mu }
1576:
1556:
1510:
1458:
1405:
1381:
1357:
1309:
1251:
1219:
1191:
1171:
1143:
1086:
1055:
989:
950:
912:
869:
837:
797:
765:
636:
604:
548:
485:
395:
384:{\displaystyle \mu }
375:
335:
298:
263:
211:
189:
161:
139:
85:
48:conservative systems
7360:Hölder's inequality
7222:of random variables
7184:Measurable function
7071:Particular measures
6660:Absolute continuity
6353:Hjorth, G. (2001).
6287:10.3934/jmd.2019024
5966: —
5804:{\displaystyle S,T}
5339:Ergodic means that
5169:topological entropy
4644:approximate entropy
4060:Further, given two
2855:
1747:dissipative systems
1051:Consider a mapping
257:probability measure
52:classical mechanics
7546:Information theory
7500:Probability theory
6825:Transverse measure
6803:Non-measurable set
6785:Locally measurable
6109:
6089:
6060:
6040:
6020:
5978:
5960:
5918:
5890:
5867:
5843:
5801:
5775:
5759:. An isomorphism
5749:
5701:
5681:
5631:
5569:
5514:
5487:
5440:
5414:
5394:
5374:
5322:
5232:
5212:
5184:
5112:
5079:
5026:
4967:
4899:
4812:
4761:
4727:
4666:
4621:
4619:
4434:
4254:
4047:
3936:
3841:
3711:
3619:
3581:
3555:
3532:
3494:
3468:
3441:
3402:
3378:
3326:
3274:
3224:
3197:
3173:
3116:
3079:
3045:
3010:
2962:
2872:
2838:
2815:
2746:
2557:
2546:
2514:) preserving map:
2479:
2456:
2404:
2384:
2364:
2293:
2273:
2150:
2136:In the case of an
2123:
2102:
2032:
2012:
1962:
1942:
1895:
1874:
1826:
1790:
1735:
1715:
1695:
1655:
1635:
1582:
1562:
1534:
1502:has the form of a
1492:
1442:
1391:
1367:
1343:
1295:
1237:
1197:
1177:
1153:
1126:
1065:
1038:
975:
936:
895:
855:
823:
783:
747:
742:
622:
587:
534:
462:
381:
353:
319:
284:
245:
195:
171:
145:
119:
76:measure-preserving
7531:Dynamical systems
7518:
7517:
7478:
7477:
7207:almost everywhere
7153:Spherical measure
7051:Strictly positive
6979:Projection-valued
6719:Almost everywhere
6692:Probability space
6542:978-0-521-88885-1
6372:10.4064/FM169-1-2
6339:10.4171/JEMS/1151
6249:978-0-521-57294-1
6112:{\displaystyle k}
6063:{\displaystyle k}
6043:{\displaystyle k}
6030:for some integer
5958:
5893:{\displaystyle U}
5704:{\displaystyle U}
5417:{\displaystyle X}
5397:{\displaystyle A}
5187:{\displaystyle T}
5135:Bernoulli process
5068:
4908:
4884:
4708:
4496:
4469:
4433:
4432: where
4209:
4182:
3856:symbolic dynamics
3336:is then called a
3078:
3020:. Then a mapping
2913:The concept of a
2758:identity function
2652:action of a group
2482:{\displaystyle T}
2407:{\displaystyle N}
2387:{\displaystyle N}
2296:{\displaystyle i}
2126:{\displaystyle X}
2035:{\displaystyle N}
1965:{\displaystyle N}
1829:{\displaystyle T}
1718:{\displaystyle T}
1667:transfer operator
1658:{\displaystyle T}
1596:; it is just the
1594:invariant measure
1565:{\displaystyle T}
1550:transfer operator
1200:{\displaystyle X}
1180:{\displaystyle X}
1167:) and also sends
721:
683:
198:{\displaystyle X}
148:{\displaystyle X}
72:probability space
36:dynamical systems
7558:
7453:Milman's reverse
7436:
7434:Lebesgue measure
7388:
7387:
6792:
6778:infimum/supremum
6699:Measurable space
6639:
6632:
6625:
6616:
6615:
6547:
6546:
6528:
6522:
6521:
6513:
6507:
6506:
6504:
6475:
6469:
6468:
6454:
6448:
6447:
6419:
6413:
6412:
6391:
6385:
6384:
6374:
6350:
6344:
6343:
6341:
6331:
6322:(8): 2605â2690.
6305:
6299:
6298:
6280:
6260:
6254:
6253:
6233:
6227:
6220:
6214:
6213:
6211:
6202:
6196:
6195:
6181:
6175:
6174:
6156:
6118:
6116:
6115:
6110:
6098:
6096:
6095:
6090:
6069:
6067:
6066:
6061:
6049:
6047:
6046:
6041:
6029:
6027:
6026:
6021:
6007:
6006:
5987:
5985:
5984:
5979:
5967:
5964:
5947:Bernoulli shifts
5927:
5925:
5924:
5919:
5917:
5916:
5899:
5897:
5896:
5891:
5876:
5874:
5873:
5868:
5866:
5865:
5852:
5850:
5849:
5844:
5827:
5826:
5810:
5808:
5807:
5802:
5784:
5782:
5781:
5776:
5758:
5756:
5755:
5750:
5733:
5732:
5710:
5708:
5707:
5702:
5690:
5688:
5687:
5682:
5671:
5670:
5640:
5638:
5637:
5632:
5630:
5629:
5617:
5616:
5601:
5600:
5578:
5576:
5575:
5570:
5562:
5561:
5546:
5545:
5523:
5521:
5520:
5515:
5513:
5512:
5496:
5494:
5493:
5488:
5474:
5469:
5461:
5449:
5447:
5446:
5441:
5423:
5421:
5420:
5415:
5403:
5401:
5400:
5395:
5383:
5381:
5380:
5375:
5358:
5357:
5331:
5329:
5328:
5323:
5306:
5295:
5284:
5258:
5257:
5241:
5239:
5238:
5233:
5221:
5219:
5218:
5213:
5211:
5193:
5191:
5190:
5185:
5146:binary expansion
5121:
5119:
5118:
5113:
5105:
5104:
5089:
5088:
5078:
5077:
5055:
5054:
5035:
5033:
5032:
5027:
5010:
5009:
4976:
4974:
4973:
4968:
4963:
4959:
4958:
4957:
4951:
4950:
4937:
4932:
4909:
4901:
4898:
4877:
4876:
4861:
4860:
4821:
4819:
4818:
4813:
4796:
4795:
4770:
4768:
4767:
4762:
4726:
4725:
4724:
4698:
4697:
4675:
4673:
4672:
4667:
4665:
4664:
4630:
4628:
4627:
4622:
4620:
4607:
4603:
4602:
4601:
4600:
4599:
4585:
4584:
4563:
4562:
4561:
4560:
4546:
4545:
4530:
4529:
4528:
4527:
4503:
4500:
4494:
4467:
4445:
4444:
4435:
4431:
4427:
4424:
4420:
4419:
4418:
4417:
4403:
4402:
4381:
4380:
4379:
4378:
4364:
4363:
4348:
4347:
4346:
4345:
4318:
4317:
4304:
4299:
4263:
4261:
4260:
4255:
4238:
4237:
4225:
4224:
4207:
4180:
4152:
4151:
4139:
4138:
4101:}, define their
4056:
4054:
4053:
4048:
4040:
4039:
4030:
4029:
4008:
4007:
3998:
3997:
3976:
3975:
3945:
3943:
3942:
3937:
3920:
3919:
3850:
3848:
3847:
3842:
3837:
3836:
3835:
3834:
3814:
3813:
3720:
3718:
3717:
3712:
3695:
3694:
3661:of the point is
3628:
3626:
3625:
3620:
3590:
3588:
3587:
3582:
3564:
3562:
3561:
3556:
3541:
3539:
3538:
3533:
3503:
3501:
3500:
3495:
3477:
3475:
3474:
3469:
3450:
3448:
3447:
3442:
3411:
3409:
3408:
3403:
3387:
3385:
3384:
3379:
3362:
3361:
3335:
3333:
3332:
3327:
3310:
3309:
3283:
3281:
3280:
3275:
3233:
3231:
3230:
3225:
3206:
3204:
3203:
3198:
3182:
3180:
3179:
3174:
3151:
3150:
3125:
3123:
3122:
3117:
3115:
3114:
3088:
3086:
3085:
3080:
3076:
3054:
3052:
3051:
3046:
3019:
3017:
3016:
3011:
2994:
2993:
2971:
2969:
2968:
2963:
2946:
2945:
2921:may be defined.
2881:
2879:
2878:
2873:
2871:
2870:
2854:
2846:
2824:
2822:
2821:
2816:
2814:
2813:
2795:
2794:
2782:
2781:
2755:
2753:
2752:
2747:
2733:
2732:
2727:
2715:
2714:
2669:parametrized by
2623:Markov processes
2580:Bernoulli scheme
2571:a rotation. See
2555:
2553:
2552:
2547:
2512:Lebesgue measure
2488:
2486:
2485:
2480:
2465:
2463:
2462:
2457:
2452:
2448:
2447:
2428:
2427:
2413:
2411:
2410:
2405:
2393:
2391:
2390:
2385:
2373:
2371:
2370:
2365:
2363:
2362:
2350:
2349:
2337:
2336:
2302:
2300:
2299:
2294:
2282:
2280:
2279:
2274:
2269:
2268:
2255:
2254:
2241:
2240:
2228:
2227:
2215:
2214:
2184:
2183:
2160:is given by the
2159:
2157:
2156:
2151:
2132:
2130:
2129:
2124:
2111:
2109:
2108:
2103:
2098:
2097:
2089:
2080:
2079:
2041:
2039:
2038:
2033:
2021:
2019:
2018:
2013:
2008:
2007:
2002:
1971:
1969:
1968:
1963:
1951:
1949:
1948:
1943:
1909:Informal example
1904:
1902:
1901:
1896:
1883:
1881:
1880:
1875:
1858:
1857:
1835:
1833:
1832:
1827:
1799:
1797:
1796:
1791:
1774:
1773:
1744:
1742:
1741:
1736:
1724:
1722:
1721:
1716:
1704:
1702:
1701:
1696:
1691:
1690:
1664:
1662:
1661:
1656:
1644:
1642:
1641:
1636:
1625:
1624:
1591:
1589:
1588:
1583:
1571:
1569:
1568:
1563:
1543:
1541:
1540:
1535:
1501:
1499:
1498:
1493:
1479:
1478:
1451:
1449:
1448:
1443:
1414:
1413:
1400:
1398:
1397:
1392:
1390:
1389:
1376:
1374:
1373:
1368:
1366:
1365:
1352:
1350:
1349:
1344:
1318:
1317:
1304:
1302:
1301:
1296:
1285:
1284:
1260:
1259:
1246:
1244:
1243:
1238:
1206:
1204:
1203:
1198:
1186:
1184:
1183:
1178:
1162:
1160:
1159:
1154:
1152:
1151:
1135:
1133:
1132:
1127:
1095:
1094:
1074:
1072:
1071:
1066:
1064:
1063:
1047:
1045:
1044:
1039:
1025:
1024:
984:
982:
981:
976:
965:
964:
945:
943:
942:
937:
904:
902:
901:
898:{\displaystyle }
896:
882:
864:
862:
861:
858:{\displaystyle }
856:
832:
830:
829:
826:{\displaystyle }
824:
816:
792:
790:
789:
786:{\displaystyle }
784:
756:
754:
753:
748:
746:
745:
736:
722:
719:
698:
684:
681:
631:
629:
628:
625:{\displaystyle }
623:
596:
594:
593:
588:
543:
541:
540:
535:
506:
505:
471:
469:
468:
463:
434:
433:
413:
412:
390:
388:
387:
382:
362:
360:
359:
354:
328:
326:
325:
320:
293:
291:
290:
285:
254:
252:
251:
246:
226:
225:
204:
202:
201:
196:
180:
178:
177:
172:
170:
169:
154:
152:
151:
146:
128:
126:
125:
120:
103:
102:
7566:
7565:
7561:
7560:
7559:
7557:
7556:
7555:
7521:
7520:
7519:
7514:
7510:Spectral theory
7490:Convex analysis
7474:
7431:
7426:
7379:
7279:
7227:in distribution
7172:
7065:
6895:Logarithmically
6834:
6790:
6773:Essential range
6707:
6648:
6643:
6556:
6554:Further reading
6551:
6550:
6543:
6529:
6525:
6514:
6510:
6476:
6472:
6455:
6451:
6436:10.2307/1968872
6420:
6416:
6409:
6392:
6388:
6351:
6347:
6306:
6302:
6261:
6257:
6250:
6234:
6230:
6221:
6217:
6209:
6203:
6199:
6182:
6178:
6171:
6157:
6150:
6145:
6126:
6121:
6104:
6101:
6100:
6078:
6075:
6074:
6055:
6052:
6051:
6035:
6032:
6031:
6002:
5998:
5996:
5993:
5992:
5973:
5970:
5969:
5965:
5962:
5951:Ornstein theory
5912:
5911:
5909:
5906:
5905:
5904:, then the set
5885:
5882:
5881:
5861:
5860:
5858:
5855:
5854:
5822:
5821:
5819:
5816:
5815:
5790:
5787:
5786:
5764:
5761:
5760:
5728:
5727:
5716:
5713:
5712:
5696:
5693:
5692:
5666:
5665:
5654:
5651:
5650:
5647:
5625:
5621:
5612:
5608:
5596:
5592:
5584:
5581:
5580:
5557:
5553:
5541:
5537:
5529:
5526:
5525:
5508:
5504:
5502:
5499:
5498:
5470:
5462:
5457:
5455:
5452:
5451:
5429:
5426:
5425:
5409:
5406:
5405:
5389:
5386:
5385:
5350:
5346:
5344:
5341:
5340:
5302:
5291:
5280:
5253:
5249:
5247:
5244:
5243:
5227:
5224:
5223:
5207:
5199:
5196:
5195:
5179:
5176:
5175:
5100:
5099:
5084:
5080:
5073:
5072:
5050:
5046:
5044:
5041:
5040:
5005:
5004:
4993:
4990:
4989:
4953:
4952:
4943:
4939:
4933:
4922:
4917:
4913:
4900:
4888:
4872:
4871:
4856:
4852:
4850:
4847:
4846:
4841:
4832:
4791:
4790:
4779:
4776:
4775:
4720:
4719:
4712:
4693:
4692:
4684:
4681:
4680:
4660:
4659:
4657:
4654:
4653:
4652:of a partition
4646:
4640:
4618:
4617:
4595:
4591:
4590:
4586:
4577:
4573:
4556:
4552:
4551:
4547:
4538:
4534:
4523:
4519:
4518:
4514:
4513:
4509:
4502:
4498:
4497:
4440:
4436:
4429:
4426:
4422:
4421:
4413:
4409:
4408:
4404:
4395:
4391:
4374:
4370:
4369:
4365:
4356:
4352:
4341:
4337:
4336:
4332:
4322:
4310:
4306:
4300:
4289:
4281:
4279:
4276:
4275:
4233:
4229:
4220:
4216:
4147:
4143:
4134:
4130:
4113:
4110:
4109:
4100:
4091:
4079:
4073:
4035:
4031:
4022:
4018:
4003:
3999:
3990:
3986:
3968:
3964:
3962:
3959:
3958:
3915:
3914:
3903:
3900:
3899:
3897:
3888:
3880:
3830:
3826:
3825:
3821:
3809:
3805:
3803:
3800:
3799:
3794:
3768:
3737:
3731:
3690:
3689:
3678:
3675:
3674:
3671:
3643:
3596:
3593:
3592:
3570:
3567:
3566:
3550:
3547:
3546:
3509:
3506:
3505:
3483:
3480:
3479:
3463:
3460:
3459:
3424:
3421:
3420:
3396:
3393:
3392:
3357:
3356:
3345:
3342:
3341:
3305:
3304:
3293:
3290:
3289:
3239:
3236:
3235:
3213:
3210:
3209:
3192:
3189:
3188:
3143:
3139:
3131:
3128:
3127:
3110:
3109:
3101:
3098:
3097:
3071:
3068:
3067:
3028:
3025:
3024:
2989:
2988:
2977:
2974:
2973:
2941:
2940:
2929:
2926:
2925:
2911:
2891:
2863:
2859:
2847:
2842:
2836:
2833:
2832:
2803:
2799:
2790:
2786:
2777:
2773:
2771:
2768:
2767:
2728:
2720:
2719:
2710:
2706:
2704:
2701:
2700:
2690:
2659:
2636:
2523:
2520:
2519:
2504:
2474:
2471:
2470:
2437:
2433:
2429:
2423:
2422:
2420:
2417:
2416:
2399:
2396:
2395:
2379:
2376:
2375:
2358:
2354:
2345:
2341:
2332:
2328:
2308:
2305:
2304:
2288:
2285:
2284:
2264:
2260:
2250:
2246:
2236:
2232:
2223:
2219:
2210:
2206:
2179:
2175:
2173:
2170:
2169:
2166:product measure
2145:
2142:
2141:
2118:
2115:
2114:
2090:
2085:
2084:
2075:
2071:
2051:
2048:
2047:
2027:
2024:
2023:
2003:
1998:
1997:
1977:
1974:
1973:
1957:
1954:
1953:
1922:
1919:
1918:
1911:
1890:
1887:
1886:
1853:
1852:
1841:
1838:
1837:
1821:
1818:
1817:
1769:
1768:
1757:
1754:
1753:
1730:
1727:
1726:
1710:
1707:
1706:
1686:
1685:
1674:
1671:
1670:
1650:
1647:
1646:
1620:
1619:
1608:
1605:
1604:
1577:
1574:
1573:
1557:
1554:
1553:
1511:
1508:
1507:
1471:
1467:
1459:
1456:
1455:
1409:
1408:
1406:
1403:
1402:
1385:
1384:
1382:
1379:
1378:
1361:
1360:
1358:
1355:
1354:
1313:
1312:
1310:
1307:
1306:
1277:
1273:
1255:
1254:
1252:
1249:
1248:
1220:
1217:
1216:
1192:
1189:
1188:
1172:
1169:
1168:
1147:
1146:
1144:
1141:
1140:
1090:
1089:
1087:
1084:
1083:
1059:
1058:
1056:
1053:
1052:
1017:
1013:
990:
987:
986:
957:
953:
951:
948:
947:
913:
910:
909:
878:
870:
867:
866:
838:
835:
834:
812:
798:
795:
794:
766:
763:
762:
741:
740:
732:
718:
703:
702:
694:
680:
667:
666:
637:
634:
633:
605:
602:
601:
549:
546:
545:
498:
494:
486:
483:
482:
479:
426:
422:
408:
407:
396:
393:
392:
376:
373:
372:
336:
333:
332:
299:
296:
295:
264:
261:
260:
221:
220:
212:
209:
208:
190:
187:
186:
165:
164:
162:
159:
158:
140:
137:
136:
98:
97:
86:
83:
82:
68:
56:non-dissipative
24:
17:
12:
11:
5:
7564:
7554:
7553:
7551:Measure theory
7548:
7543:
7538:
7533:
7516:
7515:
7513:
7512:
7507:
7502:
7497:
7492:
7486:
7484:
7480:
7479:
7476:
7475:
7473:
7472:
7467:
7462:
7457:
7456:
7455:
7445:
7439:
7437:
7428:
7427:
7425:
7424:
7419:
7417:Sard's theorem
7414:
7409:
7404:
7403:
7402:
7400:Lifting theory
7391:
7385:
7381:
7380:
7378:
7377:
7372:
7367:
7362:
7357:
7356:
7355:
7353:FubiniâTonelli
7345:
7340:
7335:
7334:
7333:
7328:
7323:
7315:
7314:
7313:
7308:
7303:
7295:
7289:
7287:
7281:
7280:
7278:
7277:
7272:
7267:
7262:
7257:
7252:
7247:
7241:
7236:
7235:
7234:
7232:in probability
7229:
7219:
7214:
7209:
7203:
7202:
7201:
7196:
7191:
7180:
7178:
7174:
7173:
7171:
7170:
7165:
7160:
7155:
7150:
7145:
7144:
7143:
7133:
7128:
7127:
7126:
7116:
7111:
7106:
7101:
7096:
7091:
7086:
7081:
7075:
7073:
7067:
7066:
7064:
7063:
7058:
7053:
7048:
7043:
7038:
7033:
7028:
7023:
7018:
7013:
7012:
7011:
7006:
7001:
6991:
6986:
6981:
6976:
6966:
6961:
6956:
6951:
6946:
6941:
6939:Locally finite
6936:
6926:
6921:
6916:
6911:
6906:
6901:
6891:
6886:
6881:
6876:
6871:
6866:
6861:
6856:
6851:
6845:
6843:
6836:
6835:
6833:
6832:
6827:
6822:
6817:
6812:
6811:
6810:
6800:
6795:
6787:
6782:
6781:
6780:
6770:
6765:
6764:
6763:
6753:
6748:
6743:
6742:
6741:
6731:
6726:
6721:
6715:
6713:
6709:
6708:
6706:
6705:
6696:
6695:
6694:
6684:
6679:
6671:
6666:
6656:
6654:
6653:Basic concepts
6650:
6649:
6646:Measure theory
6642:
6641:
6634:
6627:
6619:
6613:
6612:
6599:
6577:Lai-Sang Young
6574:
6555:
6552:
6549:
6548:
6541:
6523:
6508:
6495:(3): 337â352.
6470:
6449:
6430:(2): 332â350.
6414:
6407:
6386:
6345:
6300:
6255:
6248:
6228:
6215:
6197:
6176:
6169:
6147:
6146:
6144:
6141:
6140:
6139:
6133:
6125:
6122:
6108:
6088:
6085:
6082:
6059:
6039:
6019:
6016:
6013:
6010:
6005:
6001:
5981:{\textstyle T}
5977:
5963:(Krieger 1970)
5956:
5955:
5954:
5944:
5915:
5889:
5864:
5842:
5839:
5836:
5833:
5830:
5825:
5800:
5797:
5794:
5774:
5771:
5768:
5748:
5745:
5742:
5739:
5736:
5731:
5726:
5723:
5720:
5700:
5680:
5677:
5674:
5669:
5664:
5661:
5658:
5646:
5643:
5628:
5624:
5620:
5615:
5611:
5607:
5604:
5599:
5595:
5591:
5588:
5568:
5565:
5560:
5556:
5552:
5549:
5544:
5540:
5536:
5533:
5511:
5507:
5486:
5483:
5480:
5477:
5473:
5468:
5465:
5460:
5439:
5436:
5433:
5413:
5393:
5373:
5370:
5367:
5364:
5361:
5356:
5353:
5349:
5321:
5318:
5315:
5312:
5309:
5305:
5301:
5298:
5294:
5290:
5287:
5283:
5279:
5276:
5273:
5270:
5267:
5264:
5261:
5256:
5252:
5231:
5210:
5206:
5203:
5183:
5123:
5122:
5111:
5108:
5103:
5098:
5095:
5092:
5087:
5083:
5076:
5071:
5067:
5064:
5061:
5058:
5053:
5049:
5036:is defined as
5025:
5022:
5019:
5016:
5013:
5008:
5003:
5000:
4997:
4978:
4977:
4966:
4962:
4956:
4949:
4946:
4942:
4936:
4931:
4928:
4925:
4921:
4916:
4912:
4907:
4904:
4897:
4894:
4891:
4887:
4883:
4880:
4875:
4870:
4867:
4864:
4859:
4855:
4837:
4830:
4811:
4808:
4805:
4802:
4799:
4794:
4789:
4786:
4783:
4772:
4771:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4723:
4718:
4715:
4711:
4707:
4704:
4701:
4696:
4691:
4688:
4676:is defined as
4663:
4639:
4636:
4632:
4631:
4616:
4613:
4610:
4606:
4598:
4594:
4589:
4583:
4580:
4576:
4572:
4569:
4566:
4559:
4555:
4550:
4544:
4541:
4537:
4533:
4526:
4522:
4517:
4512:
4508:
4501:
4499:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4466:
4463:
4460:
4457:
4454:
4451:
4448:
4443:
4439:
4425:
4423:
4416:
4412:
4407:
4401:
4398:
4394:
4390:
4387:
4384:
4377:
4373:
4368:
4362:
4359:
4355:
4351:
4344:
4340:
4335:
4331:
4328:
4325:
4323:
4321:
4316:
4313:
4309:
4303:
4298:
4295:
4292:
4288:
4284:
4283:
4271:is defined as
4265:
4264:
4253:
4250:
4247:
4244:
4241:
4236:
4232:
4228:
4223:
4219:
4215:
4212:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4150:
4146:
4142:
4137:
4133:
4129:
4126:
4123:
4120:
4117:
4096:
4089:
4077:
4071:
4058:
4057:
4046:
4043:
4038:
4034:
4028:
4025:
4021:
4017:
4014:
4011:
4006:
4002:
3996:
3993:
3989:
3985:
3982:
3979:
3974:
3971:
3967:
3935:
3932:
3929:
3926:
3923:
3918:
3913:
3910:
3907:
3893:
3886:
3879:
3876:
3852:
3851:
3840:
3833:
3829:
3824:
3820:
3817:
3812:
3808:
3790:
3766:
3735:
3729:
3710:
3707:
3704:
3701:
3698:
3693:
3688:
3685:
3682:
3670:
3667:
3642:
3641:Generic points
3639:
3631:
3630:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3580:
3577:
3574:
3554:
3543:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3493:
3490:
3487:
3467:
3452:
3451:
3440:
3437:
3434:
3431:
3428:
3400:
3377:
3374:
3371:
3368:
3365:
3360:
3355:
3352:
3349:
3325:
3322:
3319:
3316:
3313:
3308:
3303:
3300:
3297:
3286:
3285:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3223:
3220:
3217:
3196:
3184:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3149:
3146:
3142:
3138:
3135:
3113:
3108:
3105:
3094:
3075:
3056:
3055:
3044:
3041:
3038:
3035:
3032:
3009:
3006:
3003:
3000:
2997:
2992:
2987:
2984:
2981:
2961:
2958:
2955:
2952:
2949:
2944:
2939:
2936:
2933:
2910:
2907:
2889:
2884:
2883:
2869:
2866:
2862:
2858:
2853:
2850:
2845:
2841:
2830:
2812:
2809:
2806:
2802:
2798:
2793:
2789:
2785:
2780:
2776:
2765:
2745:
2742:
2739:
2736:
2731:
2726:
2723:
2718:
2713:
2709:
2688:
2657:
2635:
2632:
2631:
2630:
2619:
2608:
2597:
2590:
2583:
2576:
2545:
2540:
2536:
2533:
2530:
2527:
2510:Example of a (
2503:
2500:
2478:
2455:
2451:
2446:
2443:
2440:
2436:
2432:
2426:
2403:
2383:
2361:
2357:
2353:
2348:
2344:
2340:
2335:
2331:
2327:
2324:
2321:
2318:
2315:
2312:
2292:
2272:
2267:
2263:
2258:
2253:
2249:
2244:
2239:
2235:
2231:
2226:
2222:
2218:
2213:
2209:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2182:
2178:
2149:
2140:, the measure
2122:
2101:
2096:
2093:
2088:
2083:
2078:
2074:
2070:
2067:
2064:
2061:
2058:
2055:
2031:
2011:
2006:
2001:
1996:
1993:
1990:
1987:
1984:
1981:
1961:
1952:consisting of
1941:
1938:
1935:
1932:
1929:
1926:
1910:
1907:
1894:
1873:
1870:
1867:
1864:
1861:
1856:
1851:
1848:
1845:
1825:
1814:thermalization
1789:
1786:
1783:
1780:
1777:
1772:
1767:
1764:
1761:
1734:
1714:
1694:
1689:
1684:
1681:
1678:
1654:
1634:
1631:
1628:
1623:
1618:
1615:
1612:
1581:
1572:; the measure
1561:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1491:
1488:
1485:
1482:
1477:
1474:
1470:
1466:
1463:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1412:
1388:
1364:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1316:
1294:
1291:
1288:
1283:
1280:
1276:
1272:
1269:
1266:
1263:
1258:
1236:
1233:
1230:
1227:
1224:
1196:
1176:
1150:
1137:
1136:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1093:
1062:
1037:
1034:
1031:
1028:
1023:
1020:
1016:
1012:
1009:
1006:
1003:
1000:
997:
994:
974:
971:
968:
963:
960:
956:
935:
932:
929:
926:
923:
920:
917:
894:
891:
888:
885:
881:
877:
874:
854:
851:
848:
845:
842:
822:
819:
815:
811:
808:
805:
802:
782:
779:
776:
773:
770:
757:. This is the
744:
739:
735:
731:
728:
725:
720: if
717:
714:
711:
708:
705:
704:
701:
697:
693:
690:
687:
682: if
679:
676:
673:
672:
670:
665:
662:
657:
653:
650:
647:
644:
641:
621:
618:
615:
612:
609:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
533:
530:
527:
524:
521:
518:
515:
512:
509:
504:
501:
497:
493:
490:
478:
475:
474:
473:
461:
458:
455:
452:
449:
446:
443:
440:
437:
432:
429:
425:
421:
418:
411:
406:
403:
400:
380:
352:
349:
346:
343:
340:
330:
318:
315:
312:
309:
306:
303:
283:
280:
277:
274:
271:
268:
244:
241:
238:
235:
232:
229:
224:
219:
216:
206:
194:
183:σ-algebra
168:
156:
144:
130:
129:
118:
115:
112:
109:
106:
101:
96:
93:
90:
67:
64:
40:ergodic theory
21:Equal-area map
15:
9:
6:
4:
3:
2:
7563:
7552:
7549:
7547:
7544:
7542:
7539:
7537:
7534:
7532:
7529:
7528:
7526:
7511:
7508:
7506:
7505:Real analysis
7503:
7501:
7498:
7496:
7493:
7491:
7488:
7487:
7485:
7481:
7471:
7468:
7466:
7463:
7461:
7458:
7454:
7451:
7450:
7449:
7446:
7444:
7441:
7440:
7438:
7435:
7429:
7423:
7420:
7418:
7415:
7413:
7410:
7408:
7405:
7401:
7398:
7397:
7396:
7393:
7392:
7389:
7386:
7384:Other results
7382:
7376:
7373:
7371:
7370:RadonâNikodym
7368:
7366:
7363:
7361:
7358:
7354:
7351:
7350:
7349:
7346:
7344:
7343:Fatou's lemma
7341:
7339:
7336:
7332:
7329:
7327:
7324:
7322:
7319:
7318:
7316:
7312:
7309:
7307:
7304:
7302:
7299:
7298:
7296:
7294:
7291:
7290:
7288:
7286:
7282:
7276:
7273:
7271:
7268:
7266:
7263:
7261:
7258:
7256:
7253:
7251:
7248:
7246:
7242:
7240:
7237:
7233:
7230:
7228:
7225:
7224:
7223:
7220:
7218:
7215:
7213:
7210:
7208:
7205:Convergence:
7204:
7200:
7197:
7195:
7192:
7190:
7187:
7186:
7185:
7182:
7181:
7179:
7175:
7169:
7166:
7164:
7161:
7159:
7156:
7154:
7151:
7149:
7146:
7142:
7139:
7138:
7137:
7134:
7132:
7129:
7125:
7122:
7121:
7120:
7117:
7115:
7112:
7110:
7107:
7105:
7102:
7100:
7097:
7095:
7092:
7090:
7087:
7085:
7082:
7080:
7077:
7076:
7074:
7072:
7068:
7062:
7059:
7057:
7054:
7052:
7049:
7047:
7044:
7042:
7039:
7037:
7034:
7032:
7029:
7027:
7024:
7022:
7019:
7017:
7014:
7010:
7009:Outer regular
7007:
7005:
7004:Inner regular
7002:
7000:
6999:Borel regular
6997:
6996:
6995:
6992:
6990:
6987:
6985:
6982:
6980:
6977:
6975:
6971:
6967:
6965:
6962:
6960:
6957:
6955:
6952:
6950:
6947:
6945:
6942:
6940:
6937:
6935:
6931:
6927:
6925:
6922:
6920:
6917:
6915:
6912:
6910:
6907:
6905:
6902:
6900:
6896:
6892:
6890:
6887:
6885:
6882:
6880:
6877:
6875:
6872:
6870:
6867:
6865:
6862:
6860:
6857:
6855:
6852:
6850:
6847:
6846:
6844:
6842:
6837:
6831:
6828:
6826:
6823:
6821:
6818:
6816:
6813:
6809:
6806:
6805:
6804:
6801:
6799:
6796:
6794:
6788:
6786:
6783:
6779:
6776:
6775:
6774:
6771:
6769:
6766:
6762:
6759:
6758:
6757:
6754:
6752:
6749:
6747:
6744:
6740:
6737:
6736:
6735:
6732:
6730:
6727:
6725:
6722:
6720:
6717:
6716:
6714:
6710:
6704:
6700:
6697:
6693:
6690:
6689:
6688:
6687:Measure space
6685:
6683:
6680:
6678:
6676:
6672:
6670:
6667:
6665:
6661:
6658:
6657:
6655:
6651:
6647:
6640:
6635:
6633:
6628:
6626:
6621:
6620:
6617:
6611:
6608:
6604:
6600:
6598:
6597:0-691-11338-6
6594:
6590:
6586:
6582:
6578:
6575:
6573:
6570:
6569:0-19-853390-X
6566:
6562:
6558:
6557:
6544:
6538:
6534:
6527:
6519:
6512:
6503:
6498:
6494:
6490:
6489:
6484:
6480:
6474:
6466:
6462:
6461:
6453:
6445:
6441:
6437:
6433:
6429:
6425:
6418:
6410:
6408:0-8218-2262-4
6404:
6400:
6396:
6390:
6382:
6378:
6373:
6368:
6364:
6360:
6356:
6349:
6340:
6335:
6330:
6325:
6321:
6317:
6316:
6311:
6304:
6296:
6292:
6288:
6284:
6279:
6274:
6270:
6266:
6259:
6251:
6245:
6241:
6240:
6232:
6226:
6225:
6219:
6208:
6201:
6193:
6189:
6188:
6180:
6172:
6170:0-387-95152-0
6166:
6162:
6155:
6153:
6148:
6137:
6134:
6131:
6128:
6127:
6120:
6106:
6086:
6083:
6080:
6071:
6057:
6037:
6017:
6014:
6011:
6008:
6003:
5999:
5989:
5975:
5952:
5948:
5945:
5942:
5941:
5940:
5937:
5935:
5931:
5903:
5902:weak topology
5887:
5878:
5840:
5837:
5834:
5831:
5828:
5814:
5798:
5795:
5792:
5772:
5769:
5766:
5743:
5740:
5737:
5734:
5724:
5721:
5698:
5675:
5672:
5662:
5659:
5642:
5626:
5622:
5618:
5613:
5609:
5605:
5597:
5593:
5586:
5563:
5558:
5554:
5550:
5542:
5538:
5531:
5509:
5505:
5484:
5481:
5478:
5475:
5466:
5463:
5437:
5434:
5431:
5411:
5391:
5371:
5368:
5362:
5354:
5351:
5347:
5337:
5335:
5316:
5313:
5307:
5299:
5296:
5292:
5288:
5285:
5277:
5274:
5271:
5268:
5262:
5254:
5250:
5229:
5204:
5201:
5181:
5172:
5170:
5166:
5163:If the space
5161:
5159:
5155:
5151:
5150:unit interval
5147:
5144:has a unique
5143:
5140:
5136:
5132:
5128:
5109:
5096:
5093:
5085:
5081:
5065:
5059:
5051:
5047:
5039:
5038:
5037:
5020:
5017:
5014:
5011:
5001:
4998:
4987:
4983:
4980:Finally, the
4964:
4960:
4947:
4944:
4940:
4934:
4929:
4926:
4923:
4919:
4914:
4910:
4905:
4902:
4889:
4881:
4868:
4865:
4857:
4853:
4845:
4844:
4843:
4840:
4836:
4829:
4825:
4806:
4803:
4800:
4797:
4787:
4784:
4758:
4752:
4746:
4743:
4740:
4734:
4728:
4716:
4713:
4709:
4705:
4702:
4686:
4679:
4678:
4677:
4651:
4645:
4635:
4611:
4608:
4604:
4596:
4592:
4587:
4581:
4578:
4574:
4570:
4567:
4564:
4557:
4553:
4548:
4542:
4539:
4535:
4531:
4524:
4520:
4515:
4510:
4506:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4470:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4441:
4437:
4414:
4410:
4405:
4399:
4396:
4392:
4388:
4385:
4382:
4375:
4371:
4366:
4360:
4357:
4353:
4349:
4342:
4338:
4333:
4326:
4324:
4319:
4314:
4311:
4307:
4301:
4296:
4293:
4290:
4286:
4274:
4273:
4272:
4270:
4251:
4245:
4242:
4234:
4230:
4226:
4221:
4217:
4210:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4148:
4144:
4140:
4135:
4131:
4124:
4121:
4118:
4115:
4108:
4107:
4106:
4104:
4099:
4095:
4088:
4084:
4080:
4070:
4066:
4063:
4044:
4036:
4032:
4026:
4023:
4019:
4015:
4012:
4009:
4004:
4000:
3994:
3991:
3987:
3980:
3977:
3972:
3969:
3965:
3957:
3956:
3955:
3953:
3950:-pullback of
3949:
3946:, define the
3930:
3927:
3924:
3921:
3911:
3908:
3896:
3892:
3885:
3875:
3873:
3869:
3865:
3861:
3857:
3838:
3831:
3827:
3822:
3818:
3815:
3810:
3806:
3798:
3797:
3796:
3793:
3789:
3785:
3781:
3777:
3776:symbolic name
3773:
3769:
3762:
3758:
3754:
3750:
3746:
3742:
3738:
3728:
3724:
3705:
3702:
3699:
3696:
3686:
3683:
3666:
3664:
3660:
3656:
3655:generic point
3652:
3648:
3638:
3636:
3613:
3610:
3604:
3601:
3598:
3578:
3575:
3572:
3552:
3544:
3526:
3523:
3517:
3514:
3511:
3491:
3488:
3485:
3465:
3457:
3456:
3455:
3438:
3432:
3429:
3426:
3419:
3418:
3417:
3415:
3398:
3389:
3372:
3369:
3366:
3363:
3353:
3350:
3339:
3320:
3317:
3314:
3311:
3301:
3298:
3268:
3265:
3259:
3256:
3250:
3247:
3241:
3221:
3218:
3215:
3208:
3194:
3185:
3167:
3161:
3158:
3152:
3147:
3144:
3140:
3133:
3106:
3103:
3095:
3092:
3073:
3065:
3064:
3063:
3061:
3042:
3036:
3033:
3030:
3023:
3022:
3021:
3004:
3001:
2998:
2995:
2985:
2982:
2956:
2953:
2950:
2947:
2937:
2934:
2922:
2920:
2916:
2909:Homomorphisms
2906:
2904:
2900:
2896:
2892:
2867:
2864:
2860:
2856:
2851:
2848:
2843:
2839:
2831:
2828:
2810:
2807:
2804:
2800:
2796:
2791:
2787:
2783:
2778:
2774:
2766:
2763:
2759:
2743:
2737:
2734:
2729:
2716:
2711:
2707:
2699:
2698:
2697:
2695:
2691:
2684:
2680:
2676:
2672:
2668:
2664:
2660:
2653:
2649:
2645:
2641:
2628:
2624:
2620:
2617:
2613:
2609:
2606:
2602:
2598:
2595:
2591:
2588:
2584:
2581:
2577:
2574:
2570:
2566:
2562:
2561:
2560:
2543:
2538:
2534:
2531:
2525:
2517:
2513:
2508:
2499:
2495:
2492:
2476:
2467:
2453:
2449:
2444:
2441:
2438:
2434:
2430:
2401:
2381:
2359:
2355:
2351:
2346:
2342:
2338:
2333:
2329:
2325:
2322:
2319:
2316:
2313:
2310:
2290:
2270:
2265:
2261:
2256:
2251:
2247:
2237:
2233:
2229:
2224:
2220:
2216:
2211:
2207:
2203:
2200:
2197:
2194:
2191:
2188:
2180:
2176:
2168:, in that if
2167:
2163:
2147:
2139:
2134:
2120:
2099:
2094:
2091:
2081:
2076:
2068:
2065:
2062:
2059:
2056:
2045:
2029:
2009:
2004:
1994:
1991:
1988:
1985:
1982:
1979:
1959:
1939:
1936:
1933:
1930:
1927:
1924:
1916:
1906:
1892:
1868:
1865:
1862:
1859:
1849:
1846:
1823:
1815:
1811:
1807:
1803:
1784:
1781:
1778:
1775:
1765:
1762:
1750:
1748:
1732:
1712:
1682:
1679:
1668:
1652:
1629:
1626:
1616:
1613:
1601:
1599:
1595:
1579:
1559:
1551:
1547:
1525:
1519:
1513:
1505:
1483:
1475:
1472:
1468:
1461:
1453:
1439:
1433:
1427:
1424:
1418:
1337:
1331:
1328:
1322:
1289:
1281:
1278:
1274:
1270:
1264:
1234:
1228:
1225:
1222:
1214:
1210:
1194:
1174:
1166:
1120:
1114:
1105:
1099:
1096:
1082:
1081:
1080:
1078:
1049:
1029:
1021:
1018:
1014:
1007:
1004:
998:
992:
969:
961:
958:
954:
930:
927:
924:
918:
915:
906:
889:
886:
883:
879:
875:
849:
846:
843:
817:
813:
809:
806:
803:
777:
774:
771:
760:
759:Bernoulli map
737:
733:
729:
726:
723:
715:
712:
709:
706:
699:
695:
691:
688:
685:
677:
674:
668:
663:
660:
655:
651:
648:
645:
642:
639:
616:
613:
610:
598:
581:
575:
572:
563:
557:
551:
528:
522:
519:
510:
502:
499:
495:
488:
456:
450:
447:
438:
430:
427:
423:
416:
404:
401:
378:
370:
366:
350:
344:
341:
338:
331:
316:
313:
301:
281:
278:
272:
266:
258:
239:
236:
233:
217:
214:
207:
192:
184:
157:
142:
135:
134:
133:
113:
110:
107:
104:
94:
91:
81:
80:
79:
77:
73:
63:
61:
57:
53:
49:
45:
41:
37:
33:
29:
22:
7285:Main results
7021:Set function
6949:Metric outer
6904:Decomposable
6761:Cylinder set
6674:
6609:
6607:PDF-Document
6602:
6588:
6571:
6560:
6532:
6526:
6517:
6511:
6492:
6486:
6479:Ornstein, D.
6473:
6464:
6458:
6452:
6427:
6423:
6417:
6398:
6395:Ornstein, D.
6389:
6365:(1): 51â84.
6362:
6358:
6348:
6319:
6313:
6303:
6268:
6264:
6258:
6238:
6231:
6222:
6218:
6200:
6191:
6185:
6179:
6163:. Springer.
6160:
6072:
5990:
5957:
5938:
5879:
5648:
5338:
5334:logistic map
5173:
5164:
5162:
5157:
5153:
5139:almost every
5124:
4985:
4981:
4979:
4838:
4834:
4827:
4823:
4773:
4647:
4633:
4268:
4266:
4097:
4093:
4086:
4082:
4075:
4068:
4064:
4059:
3951:
3947:
3894:
3890:
3883:
3881:
3871:
3867:
3863:
3862:is called a
3859:
3853:
3795:} such that
3791:
3787:
3783:
3779:
3775:
3771:
3764:
3760:
3756:
3752:
3748:
3744:
3733:
3726:
3722:
3672:
3654:
3653:is called a
3650:
3646:
3644:
3632:
3565:-almost all
3478:-almost all
3453:
3413:
3390:
3337:
3287:
3059:
3057:
2923:
2915:homomorphism
2912:
2902:
2898:
2894:
2887:
2885:
2827:well-defined
2761:
2693:
2686:
2682:
2678:
2674:
2670:
2666:
2662:
2655:
2639:
2637:
2568:
2558:
2515:
2496:
2468:
2374:, then, for
2135:
2044:single point
2043:
1912:
1751:
1602:
1454:
1209:conservative
1138:
1050:
907:
632:, and a map
599:
480:
371:the measure
131:
69:
31:
25:
7245:compact set
7212:of measures
7148:Pushforward
7141:Projections
7131:Logarithmic
6974:Probability
6964:Pre-measure
6746:Borel space
6664:of measures
6271:: 345â423.
6070:generator.
5811:defines an
5142:real number
5131:Yakov Sinai
3288:The system
3207:-almost all
2919:isomorphism
2646:(or even a
2565:unit circle
1504:pushforward
1247:by writing
28:mathematics
7525:Categories
7217:in measure
6944:Maximising
6914:Equivalent
6808:Vitali set
6467:: 797â800.
6359:Fund. Math
6329:1705.04414
6278:1703.07093
6194:: 768â771.
6143:References
5125:where the
4642:See also:
4103:refinement
4062:partitions
3759:, clearly
3721:, and let
3591:, one has
3504:, one has
3234:, one has
3126:, one has
3091:measurable
2164:. It is a
1810:turbulence
1506:, whereas
1213:surjective
1165:Borel sets
1077:power sets
477:Discussion
365:measurable
259:, so that
66:Definition
7331:Maharam's
7301:Dominated
7114:Intensity
7109:Hausdorff
7016:Saturated
6934:Invariant
6839:Types of
6798:Ï-algebra
6768:đ-system
6734:Borel set
6729:Baire set
6295:119128525
6084:
6015:
6009:≤
5953:for more.
5930:Borel set
5928:is not a
5835:×
5829:⊂
5770:∼
5738:μ
5676:μ
5606:∩
5567:∅
5551:∩
5485:ϵ
5476:≥
5432:ϵ
5352:−
5308:μ
5278:
5272:∫
5255:μ
5230:μ
5205:⊂
5086:μ
5052:μ
5021:μ
4945:−
4920:⋁
4896:∞
4893:→
4858:μ
4807:μ
4747:μ
4744:
4729:μ
4717:∈
4710:∑
4706:−
4579:−
4571:∩
4568:⋯
4565:∩
4540:−
4532:∩
4507:μ
4483:…
4471:ℓ
4456:…
4442:ℓ
4397:−
4389:∩
4386:⋯
4383:∩
4358:−
4350:∩
4312:−
4287:⋁
4227:∩
4211:μ
4196:…
4169:…
4154:∣
4141:∩
4119:∨
4024:−
4013:…
3992:−
3970:−
3931:μ
3864:generator
3819:∈
3741:partition
3706:μ
3611:ψ
3605:φ
3576:∈
3553:ν
3524:φ
3518:ψ
3489:∈
3466:μ
3436:→
3427:ψ
3399:φ
3367:μ
3315:ν
3266:φ
3242:φ
3219:∈
3195:μ
3162:ν
3145:−
3141:φ
3134:μ
3107:∈
3096:For each
3074:φ
3040:→
3031:φ
2999:ν
2951:μ
2865:−
2849:−
2784:∘
2741:→
2601:base flow
2529:↦
2439:−
2148:μ
2138:ideal gas
2082:×
2066:×
2060:×
1995:×
1989:×
1983:×
1934:×
1928:×
1893:μ
1863:μ
1779:μ
1733:μ
1630:μ
1580:μ
1514:μ
1473:−
1462:μ
1279:−
1232:→
1112:→
1019:−
1008:μ
993:μ
959:−
919:⊂
713:−
576:μ
552:μ
523:μ
500:−
489:μ
451:μ
428:−
417:μ
405:∈
399:∀
379:μ
369:preserves
348:→
308:∅
302:μ
267:μ
228:→
215:μ
155:is a set,
108:μ
7348:Fubini's
7338:Egorov's
7306:Monotone
7265:variable
7243:Random:
7194:Strongly
7119:Lebesgue
7104:Harmonic
7094:Gaussian
7079:Counting
7046:Spectral
7041:Singular
7031:s-finite
7026:Ï-finite
6909:Discrete
6884:Complete
6841:Measures
6815:Null set
6703:function
6481:(1970).
6381:55619325
6124:See also
5467:′
5384:implies
5127:supremum
3645:A point
3635:category
3391:The map
3066:The map
2661: :
2502:Examples
1669:, fixes
1546:pullback
391:, i.e.,
7536:Entropy
7260:process
7255:measure
7250:element
7189:Bochner
7163:Trivial
7158:Tangent
7136:Product
6994:Regular
6972:)
6959:Perfect
6932:)
6897:)
6889:Content
6879:Complex
6820:Support
6793:-system
6682:Measure
6589:Entropy
6444:1968872
6426:. (2).
4833:, ...,
4650:entropy
4092:, ...,
4074:, ...,
3889:, ...,
3739:} be a
3732:, ...,
3657:if the
2917:and an
2133:above.
7326:Jordan
7311:Vitali
7270:vector
7199:Weakly
7061:Vector
7036:Signed
6989:Random
6930:Quasi-
6919:Finite
6899:Convex
6859:Banach
6849:Atomic
6677:spaces
6662:
6595:
6567:
6539:
6442:
6405:
6379:
6293:
6246:
6167:
5961:
5222:, and
4495:
4468:
4208:
4181:
4081:} and
3412:is an
3338:factor
3077:
2756:, the
2644:monoid
2625:, the
2567:, and
1806:mixing
294:, and
74:and a
38:, and
7168:Young
7089:Euler
7084:Dirac
7056:Tight
6984:Radon
6954:Outer
6924:Inner
6874:Brown
6869:Borel
6864:Besov
6854:Baire
6440:JSTOR
6377:S2CID
6324:arXiv
6291:S2CID
6273:arXiv
6210:(PDF)
3747:into
3659:orbit
3058:is a
2681:, or
2648:group
2614:) by
2603:of a
363:is a
255:is a
185:over
181:is a
7432:For
7321:Hahn
7177:Maps
7099:Haar
6970:Sub-
6724:Atom
6712:Sets
6593:ISBN
6565:ISBN
6537:ISBN
6403:ISBN
6244:ISBN
6165:ISBN
5435:>
4648:The
4609:>
4243:>
3545:for
3458:for
3186:For
2972:and
2897:for
2677:(or
2599:the
2585:the
2578:the
1913:The
1705:and
1215:map
727:>
689:<
30:, a
6581:pdf
6497:doi
6465:147
6432:doi
6367:doi
6363:169
6334:doi
6283:doi
6192:124
5991:If
5579:or
5174:If
5070:sup
4984:or
4886:lim
4826:= {
4741:log
4105:as
4085:= {
4067:= {
3954:as
3866:or
3778:of
3743:of
3725:= {
3340:of
3089:is
2760:on
2539:mod
1187:to
1075:of
656:mod
26:In
7527::
6585:ps
6583:;
6491:.
6485:.
6463:.
6438:.
6428:43
6375:.
6361:.
6357:.
6332:.
6320:24
6318:.
6312:.
6289:.
6281:.
6269:15
6267:.
6190:.
6151:^
6081:ln
6012:ln
5641:.
5450:,
5336:.
5275:ln
5160:.
3772:Tx
3755:â
3649:â
3388:.
2905:.
2901:â
2893:=
2673:â
2665:â
2544:1.
1812:,
1808:,
1452:.
1079::
1048:.
62:.
6968:(
6928:(
6893:(
6791:Ï
6701:/
6675:L
6638:e
6631:t
6624:v
6545:.
6505:.
6499::
6493:4
6446:.
6434::
6411:.
6383:.
6369::
6342:.
6336::
6326::
6297:.
6285::
6275::
6252:.
6212:.
6173:.
6107:k
6087:k
6058:k
6038:k
6018:k
6004:T
6000:h
5976:T
5914:R
5888:U
5863:R
5841:.
5838:U
5832:U
5824:R
5799:T
5796:,
5793:S
5773:T
5767:S
5747:)
5744:T
5741:,
5735:,
5730:B
5725:,
5722:X
5719:(
5699:U
5679:)
5673:,
5668:B
5663:,
5660:X
5657:(
5627:i
5623:I
5619:=
5614:i
5610:I
5603:)
5598:i
5594:I
5590:(
5587:T
5564:=
5559:i
5555:I
5548:)
5543:i
5539:I
5535:(
5532:T
5510:i
5506:I
5482:+
5479:1
5472:|
5464:T
5459:|
5438:0
5412:X
5392:A
5372:A
5369:=
5366:)
5363:A
5360:(
5355:1
5348:T
5320:)
5317:x
5314:d
5311:(
5304:|
5300:x
5297:d
5293:/
5289:T
5286:d
5282:|
5269:=
5266:)
5263:T
5260:(
5251:h
5209:R
5202:X
5182:T
5165:X
5158:x
5154:x
5110:.
5107:)
5102:Q
5097:,
5094:T
5091:(
5082:h
5075:Q
5066:=
5063:)
5060:T
5057:(
5048:h
5024:)
5018:,
5015:T
5012:,
5007:B
5002:,
4999:X
4996:(
4965:.
4961:)
4955:Q
4948:n
4941:T
4935:N
4930:0
4927:=
4924:n
4915:(
4911:H
4906:N
4903:1
4890:N
4882:=
4879:)
4874:Q
4869:,
4866:T
4863:(
4854:h
4839:k
4835:Q
4831:1
4828:Q
4824:Q
4810:)
4804:,
4801:T
4798:,
4793:B
4788:,
4785:X
4782:(
4759:.
4756:)
4753:Q
4750:(
4738:)
4735:Q
4732:(
4722:Q
4714:Q
4703:=
4700:)
4695:Q
4690:(
4687:H
4662:Q
4615:}
4612:0
4605:)
4597:N
4593:i
4588:Q
4582:N
4575:T
4558:1
4554:i
4549:Q
4543:1
4536:T
4525:0
4521:i
4516:Q
4511:(
4492:,
4489:N
4486:,
4480:,
4477:0
4474:=
4465:,
4462:k
4459:,
4453:,
4450:1
4447:=
4438:i
4415:N
4411:i
4406:Q
4400:N
4393:T
4376:1
4372:i
4367:Q
4361:1
4354:T
4343:0
4339:i
4334:Q
4330:{
4327:=
4320:Q
4315:n
4308:T
4302:N
4297:0
4294:=
4291:n
4252:.
4249:}
4246:0
4240:)
4235:j
4231:R
4222:i
4218:Q
4214:(
4205:,
4202:m
4199:,
4193:,
4190:1
4187:=
4184:j
4178:,
4175:k
4172:,
4166:,
4163:1
4160:=
4157:i
4149:j
4145:R
4136:i
4132:Q
4128:{
4125:=
4122:R
4116:Q
4098:m
4094:R
4090:1
4087:R
4083:R
4078:k
4076:Q
4072:1
4069:Q
4065:Q
4045:.
4042:}
4037:k
4033:Q
4027:1
4020:T
4016:,
4010:,
4005:1
4001:Q
3995:1
3988:T
3984:{
3981:=
3978:Q
3973:1
3966:T
3952:Q
3948:T
3934:)
3928:,
3925:T
3922:,
3917:B
3912:,
3909:X
3906:(
3895:k
3891:Q
3887:1
3884:Q
3872:x
3860:Q
3839:.
3832:n
3828:a
3823:Q
3816:x
3811:n
3807:T
3792:n
3788:a
3784:Q
3780:x
3767:i
3765:Q
3761:x
3757:X
3753:x
3749:k
3745:X
3736:k
3734:Q
3730:1
3727:Q
3723:Q
3709:)
3703:,
3700:T
3697:,
3692:B
3687:,
3684:X
3681:(
3651:X
3647:x
3629:.
3617:)
3614:y
3608:(
3602:=
3599:y
3579:Y
3573:y
3542:;
3530:)
3527:x
3521:(
3515:=
3512:x
3492:X
3486:x
3439:X
3433:Y
3430::
3376:)
3373:T
3370:,
3364:,
3359:A
3354:,
3351:X
3348:(
3324:)
3321:S
3318:,
3312:,
3307:B
3302:,
3299:Y
3296:(
3284:.
3272:)
3269:x
3263:(
3260:S
3257:=
3254:)
3251:x
3248:T
3245:(
3222:X
3216:x
3183:.
3171:)
3168:B
3165:(
3159:=
3156:)
3153:B
3148:1
3137:(
3112:B
3104:B
3093:.
3043:Y
3037:X
3034::
3008:)
3005:S
3002:,
2996:,
2991:B
2986:,
2983:Y
2980:(
2960:)
2957:T
2954:,
2948:,
2943:A
2938:,
2935:X
2932:(
2903:N
2899:s
2895:T
2890:s
2888:T
2868:s
2861:T
2857:=
2852:1
2844:s
2840:T
2829:;
2811:s
2808:+
2805:t
2801:T
2797:=
2792:t
2788:T
2779:s
2775:T
2764:;
2762:X
2744:X
2738:X
2735::
2730:X
2725:d
2722:i
2717:=
2712:0
2708:T
2694:T
2689:s
2687:T
2683:N
2679:R
2675:Z
2671:s
2667:X
2663:X
2658:s
2656:T
2640:T
2618:;
2607:;
2596:;
2589:;
2582:;
2575:;
2569:T
2535:x
2532:2
2526:x
2516:T
2477:T
2454:.
2450:)
2445:N
2442:3
2435:2
2431:(
2425:O
2402:N
2382:N
2360:z
2356:v
2352:,
2347:y
2343:v
2339:,
2334:x
2330:v
2326:,
2323:z
2320:,
2317:y
2314:,
2311:x
2291:i
2271:p
2266:3
2262:d
2257:x
2252:3
2248:d
2243:)
2238:z
2234:v
2230:,
2225:y
2221:v
2217:,
2212:x
2208:v
2204:,
2201:z
2198:,
2195:y
2192:,
2189:x
2186:(
2181:i
2177:p
2121:X
2100:.
2095:N
2092:3
2087:R
2077:N
2073:)
2069:h
2063:l
2057:w
2054:(
2030:N
2010:.
2005:3
2000:R
1992:h
1986:l
1980:w
1960:N
1940:,
1937:h
1931:l
1925:w
1872:)
1869:T
1866:,
1860:,
1855:B
1850:,
1847:X
1844:(
1824:T
1788:)
1785:T
1782:,
1776:,
1771:B
1766:,
1763:X
1760:(
1713:T
1693:)
1688:B
1683:,
1680:X
1677:(
1653:T
1633:)
1627:,
1622:B
1617:,
1614:X
1611:(
1560:T
1532:)
1529:)
1526:A
1523:(
1520:T
1517:(
1490:)
1487:)
1484:A
1481:(
1476:1
1469:T
1465:(
1440:;
1437:)
1434:A
1431:(
1428:T
1425:=
1422:)
1419:A
1416:(
1411:T
1387:T
1363:T
1341:)
1338:A
1335:(
1332:T
1329:=
1326:)
1323:A
1320:(
1315:T
1293:)
1290:A
1287:(
1282:1
1275:T
1271:=
1268:)
1265:A
1262:(
1257:T
1235:X
1229:X
1226::
1223:T
1195:X
1175:X
1149:T
1124:)
1121:X
1118:(
1115:P
1109:)
1106:X
1103:(
1100:P
1097::
1092:T
1061:T
1036:)
1033:)
1030:A
1027:(
1022:1
1015:T
1011:(
1005:=
1002:)
999:A
996:(
973:)
970:A
967:(
962:1
955:T
934:]
931:1
928:,
925:0
922:[
916:A
893:]
890:1
887:,
884:2
880:/
876:1
873:[
853:]
850:1
847:,
844:0
841:[
821:]
818:2
814:/
810:1
807:,
804:0
801:[
781:]
778:1
775:,
772:0
769:[
738:2
734:/
730:1
724:x
716:1
710:x
707:2
700:2
696:/
692:1
686:x
678:x
675:2
669:{
664:=
661:1
652:x
649:2
646:=
643:x
640:T
620:]
617:1
614:,
611:0
608:[
585:)
582:A
579:(
573:=
570:)
567:)
564:A
561:(
558:T
555:(
532:)
529:A
526:(
520:=
517:)
514:)
511:A
508:(
503:1
496:T
492:(
472:.
460:)
457:A
454:(
448:=
445:)
442:)
439:A
436:(
431:1
424:T
420:(
410:B
402:A
351:X
345:X
342::
339:T
329:,
317:0
314:=
311:)
305:(
282:1
279:=
276:)
273:X
270:(
243:]
240:1
237:,
234:0
231:[
223:B
218::
205:,
193:X
167:B
143:X
117:)
114:T
111:,
105:,
100:B
95:,
92:X
89:(
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.