426:
2970:
is independent of the partition (or rather, there are isomorphisms connecting the symbolic dynamics of different partitions, leaving the measure invariant), and so such systems can have a well-defined entropy independent of the partition.
997:
2957:
2167:
2301:
1932:
1769:
1695:
815:
506:
2708:
273:
1266:
2998:
can be judged to be isomorphic to
Bernoulli schemes. The result was surprising, as many systems previously believed to be unrelated proved to be isomorphic. These include all finite
1169:
1447:
1391:
284:
2761:
2085:
593:
210:
1558:
2611:
1107:
2803:
2370:
2847:
1798:
1328:
624:
536:
1501:
2195:
2438:
1033:
2470:
2405:
1624:
1591:
2497:
145:
2553:
2028:
826:
3348:
3264:
2221:
1995:
1975:
1955:
1842:
1822:
2529:
693:
2093:
2229:
1850:
1700:
2855:
1629:
701:
444:
3066:
17:
3228:
2616:
These generalizations are also commonly called
Bernoulli schemes, as they still share most properties with the finite case.
3384:
2995:
1175:
2994:
The
Ornstein isomorphism theorem is in fact considerably deeper: it provides a simple criterion by which many different
2637:
3069:
to a
Bernoulli shift; in the case of zero entropy, if it is Kakutani-equivalent to an irrational rotation of a circle.
2172:
This works because the countable direct product of a standard probability space is again a standard probability space.
225:
3450:
3129:
3038:
2718:
1117:
1192:
3050:
2984:
3196:
421:{\displaystyle X=\{x=(\ldots ,x_{-1},x_{0},x_{1},\ldots ):x_{k}\in \{1,\ldots ,N\}\;\forall k\in \mathbb {Z} \}.}
1126:
1396:
1340:
3270:
3206:
2999:
2724:
2048:
556:
161:
70:, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the
2980:
83:
1506:
3201:
2565:
3315:
3478:
3329:
3042:
2307:
2043:
2030:
and so is not quite a true metric; despite this, it is commonly called a "distance" in the literature.
1054:
2315:
1776:
1306:
605:
517:
3120:
Michael S. Keane, "Ergodic theory and subshifts of finite type", (1991), appearing as
Chapter 2 in
3003:
1460:
106:
2178:
3473:
3166:
2988:
2777:
2812:
2410:
3468:
3371:
3124:, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991).
3088:
1005:
2443:
2378:
1596:
1563:
1288:
432:
992:{\displaystyle \mu \left(\right)=\mathrm {Pr} (X_{0}=x_{0},X_{1}=x_{1},\ldots ,X_{n}=x_{n})}
3078:
3015:
2475:
1303:
provides a natural metric on a
Bernoulli scheme. Another important metric is the so-called
123:
2538:
2004:
8:
1998:
3422:
3333:
3249:
2772:
2628:
2532:
2206:
2042:, rather than from the finite base space. Thus, one may take the base space to be any
1980:
1960:
1940:
1827:
1807:
102:
87:
2502:
633:
3446:
3224:
3180:
3161:
3125:
2967:
2714:
1276:
47:
43:
3021:
For the generalized case, the
Ornstein isomorphism theorem still holds if the group
3393:
3279:
3175:
2963:
2768:
1300:
1284:
1038:
71:
51:
3157:
3054:
109:
67:
3375:
3026:
2201:
2039:
1042:
543:
79:
55:
2713:
This may be seen as resulting from the general definition of the entropy of a
2162:{\displaystyle (X,{\mathcal {A}},\mu )=(Y,{\mathcal {B}},\nu )^{\mathbb {Z} }}
1112:
Since the outcomes are independent, the shift preserves the measure, and thus
3462:
599:
512:
99:
3083:
3007:
2556:
1280:
627:
216:
3413:
Bowen, Lewis (2012). "Every countably infinite group is almost
Ornstein".
46:
to more than two possible outcomes. Bernoulli schemes appear naturally in
27:
Generalization of the
Bernoulli process to more than two possible outcomes
3011:
2987:. The result is sharp, in that very similar, non-scheme systems, such as
2962:
In general, this entropy will depend on the partition; however, for many
2038:
Most of the properties of the
Bernoulli scheme follow from the countable
31:
820:
The equivalent expression, using the notation of probability theory, is
3398:
3379:
3284:
3245:
1560:
understood to be totally ordered. That is, each individual subsequence
63:
3298:
Ya.G. Sinai, (1959) "On the Notion of Entropy of a Dynamical System",
2624:
2198:
1037:
The Bernoulli scheme, as any stochastic process, may be viewed as a
2296:{\displaystyle (X,{\mathcal {A}},\mu )=(Y,{\mathcal {B}},\nu )^{G}}
59:
3427:
2767:
a base space which is not countable), one typically considers the
1279:. The Bernoulli shift can be understood as a special case of the
3380:"Entropy and isomorphism theorems for actions of amenable groups"
1927:{\displaystyle {\overline {f}}(A,B)=1-{\frac {2\sup |M|}{m+n}}}
1764:{\displaystyle 1\leq j_{1}<j_{2}<\cdots <j_{r}\leq n.}
2952:{\displaystyle H_{Y'}=-\sum _{y'\in Y'}\nu (y')\log \nu (y').}
1690:{\displaystyle 1\leq i_{1}<i_{2}<\cdots <i_{r}\leq m}
542:
is the product sigma algebra; that is, it is the (countable)
2983:
states that two Bernoulli schemes with the same entropy are
810:{\displaystyle \mu \left(\right)=\prod _{i=0}^{n}p_{x_{i}}}
501:{\displaystyle \mu =\{p_{1},\ldots ,p_{N}\}^{\mathbb {Z} }}
2306:
For this last case, the shift operator is replaced by the
2175:
As a further generalization, one may replace the integers
3162:"Bernoulli shifts with the same entropy are isomorphic"
3122:
Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces
546:
of the σ-algebras of the finite set {1, ...,
86:
shows that Bernoulli shifts are isomorphic when their
3336:
3252:
2858:
2815:
2780:
2727:
2640:
2568:
2541:
2505:
2478:
2446:
2413:
2381:
2318:
2232:
2209:
2181:
2096:
2051:
2007:
1983:
1963:
1943:
1853:
1830:
1810:
1779:
1703:
1632:
1599:
1566:
1509:
1463:
1399:
1343:
1309:
1195:
1129:
1057:
1008:
829:
704:
636:
608:
559:
520:
447:
287:
228:
164:
126:
82:, which may be used to study Bernoulli schemes. The
1937:where the supremum is being taken over all matches
3342:
3258:
2951:
2841:
2797:
2755:
2703:{\displaystyle H=-\sum _{i=1}^{N}p_{i}\log p_{i}.}
2702:
2605:
2547:
2523:
2491:
2464:
2432:
2399:
2364:
2295:
2215:
2189:
2161:
2079:
2022:
1989:
1969:
1949:
1926:
1836:
1816:
1792:
1763:
1689:
1618:
1585:
1552:
1495:
1441:
1385:
1322:
1260:
1163:
1101:
1027:
991:
809:
687:
618:
587:
530:
500:
420:
267:
204:
139:
3354:Transactions of the American Mathematical Society
3018:: these are all isomorphic to Bernoulli schemes.
1503:of indexes into the string, i.e. pairs such that
3460:
3065:A system is termed "loosely Bernoulli" if it is
1894:
268:{\displaystyle X=\{1,\ldots ,N\}^{\mathbb {Z} }}
2974:
2717:of probability spaces, which follows from the
2559:, which is invariant under the group action:
1287:are one, the corresponding graph thus being a
1261:{\displaystyle BS(p)=BS(p_{1},\ldots ,p_{N}).}
3370:
3194:
2499:can be understood to be the set of functions
1022:
1009:
487:
454:
412:
394:
376:
294:
254:
235:
54:. Many important dynamical systems (such as
3142:Chaos, scattering and statistical mechanics
116:distinct possible values, with the outcome
2771:. So, for example, if one has a countable
397:
3426:
3397:
3283:
3266:-automorphisms and a problem of Kakutani"
3179:
3032:
2602:
2183:
2153:
1164:{\displaystyle (X,{\mathcal {A}},\mu ,T)}
492:
408:
259:
50:, and are thus important in the study of
3156:
2721:. For the case of a general base space
1442:{\displaystyle B=b_{1}b_{2}\cdots b_{n}}
1386:{\displaystyle A=a_{1}a_{2}\cdots a_{m}}
3327:
3243:
2756:{\displaystyle (Y,{\mathcal {B}},\nu )}
2080:{\displaystyle (Y,{\mathcal {B}},\nu )}
588:{\displaystyle (X,{\mathcal {A}},\mu )}
205:{\displaystyle \sum _{i=1}^{N}p_{i}=1.}
14:
3461:
3300:Doklady of Russian Academy of Sciences
3218:
1294:
3412:
3152:
3150:
2087:, and define the Bernoulli scheme as
3060:
2996:measure-preserving dynamical systems
1553:{\displaystyle a_{i_{k}}=b_{j_{k}},}
1330:metric, defined via a supremum over
2606:{\displaystyle \mu (gx)=\mu (x).\,}
1176:measure-preserving dynamical system
24:
3316:Metric Entropy of Dynamical System
3147:
3144:(1998), Cambridge University press
2739:
2631:of a Bernoulli scheme is given by
2272:
2244:
2136:
2108:
2063:
2033:
1141:
898:
895:
611:
571:
523:
398:
25:
3490:
3443:An Introduction to Ergodic Theory
3039:measure-preserving transformation
2719:asymptotic equipartition property
1275:= 2 Bernoulli scheme is called a
1118:measure-preserving transformation
1102:{\displaystyle T(x_{k})=x_{k+1}.}
2849:, one may define the entropy as
2365:{\displaystyle gx(f)=x(g^{-1}f)}
3435:
3406:
3364:
3321:
3000:stationary stochastic processes
1793:{\displaystyle {\overline {f}}}
1323:{\displaystyle {\overline {f}}}
151: = 1, ...,
3385:Journal d'Analyse Mathématique
3308:
3292:
3237:
3212:
3197:"Ornstein isomorphism theorem"
3188:
3134:
3114:
3109:The theory of Bernoulli shifts
3101:
2943:
2932:
2920:
2909:
2830:
2819:
2750:
2728:
2596:
2590:
2581:
2572:
2518:
2512:
2506:
2456:
2359:
2340:
2331:
2325:
2284:
2261:
2255:
2233:
2148:
2125:
2119:
2097:
2074:
2052:
1906:
1898:
1876:
1864:
1613:
1600:
1580:
1567:
1490:
1464:
1252:
1220:
1208:
1202:
1158:
1130:
1074:
1061:
986:
902:
883:
838:
758:
713:
682:
637:
619:{\displaystyle {\mathcal {A}}}
582:
560:
531:{\displaystyle {\mathcal {A}}}
357:
303:
13:
1:
3328:Hoffman, Christopher (1999).
3271:Israel Journal of Mathematics
3111:, Univ. Chicago Press (1973)
3094:
3045:(Lebesgue space) is called a
2991:, do not have this property.
2619:
1496:{\displaystyle (i_{k},j_{k})}
1449:be two strings of symbols. A
93:
3181:10.1016/0001-8708(70)90029-0
2981:Ornstein isomorphism theorem
2975:Ornstein isomorphism theorem
2190:{\displaystyle \mathbb {Z} }
1859:
1785:
1315:
84:Ornstein isomorphism theorem
7:
3202:Encyclopedia of Mathematics
3072:
2798:{\displaystyle Y'\subset Y}
1283:, where all entries in the
120:occurring with probability
62:that is the product of the
42:is a generalization of the
10:
3495:
3043:standard probability space
2966:, it is the case that the
2842:{\displaystyle \nu (Y')=1}
2433:{\displaystyle x\in Y^{G}}
2044:standard probability space
2440:understood as a function
1186:. It is often denoted by
1028:{\displaystyle \{X_{k}\}}
1002:for the random variables
3415:Contemporary Mathematics
3025:is a countably infinite
3004:subshifts of finite type
2989:Kolmogorov automorphisms
2465:{\displaystyle x:G\to Y}
2400:{\displaystyle f,g\in G}
1041:by endowing it with the
98:A Bernoulli scheme is a
3350:Counterexample Machine"
3244:Feldman, Jacob (1976).
3195:D.S. Ornstein (2001) ,
3167:Advances in Mathematics
1619:{\displaystyle (j_{k})}
1586:{\displaystyle (i_{k})}
630:. Given a cylinder set
78:is in reference to the
3344:
3314:Ya. G. Sinai, (2007) "
3260:
3219:Klenke, Achim (2006).
3089:Hidden Bernoulli model
3047:Bernoulli automorphism
3033:Bernoulli automorphism
2953:
2843:
2799:
2757:
2704:
2670:
2627:demonstrated that the
2607:
2549:
2525:
2493:
2466:
2434:
2401:
2366:
2297:
2217:
2191:
2163:
2081:
2024:
1991:
1971:
1951:
1928:
1838:
1818:
1794:
1765:
1691:
1620:
1587:
1554:
1497:
1443:
1387:
1324:
1262:
1165:
1103:
1029:
993:
811:
789:
689:
620:
589:
550:}. Thus, the triplet
532:
502:
422:
269:
219:is usually denoted as
206:
185:
141:
18:Bernoulli automorphism
3441:Peter Walters (1982)
3345:
3261:
2954:
2844:
2800:
2758:
2705:
2650:
2608:
2550:
2526:
2494:
2492:{\displaystyle Y^{G}}
2467:
2435:
2402:
2367:
2298:
2218:
2192:
2164:
2082:
2025:
1997:. This satisfies the
1992:
1972:
1952:
1929:
1839:
1819:
1795:
1766:
1692:
1621:
1588:
1555:
1498:
1444:
1388:
1325:
1263:
1166:
1104:
1030:
994:
812:
769:
690:
621:
590:
533:
503:
423:
270:
207:
165:
142:
140:{\displaystyle p_{i}}
3334:
3250:
3079:Shift of finite type
2856:
2813:
2778:
2725:
2638:
2566:
2548:{\displaystyle \mu }
2539:
2503:
2476:
2472:(any direct product
2444:
2411:
2379:
2316:
2230:
2207:
2179:
2094:
2049:
2023:{\displaystyle m=n,}
2005:
1981:
1961:
1941:
1851:
1828:
1808:
1777:
1701:
1630:
1597:
1564:
1507:
1461:
1397:
1341:
1307:
1193:
1127:
1055:
1006:
827:
702:
634:
606:
557:
518:
445:
285:
226:
162:
124:
3445:, Springer-Verlag,
3372:Ornstein, Donald S.
3223:. Springer-Verlag.
3067:Kakutani-equivalent
2375:for group elements
1999:triangle inequality
1295:Matches and metrics
278:as a shorthand for
112:may take on one of
3399:10.1007/BF02790325
3340:
3285:10.1007/BF02761426
3256:
3221:Probability Theory
2949:
2905:
2839:
2795:
2753:
2700:
2629:Kolmogorov entropy
2603:
2545:
2533:exponential object
2521:
2489:
2462:
2430:
2397:
2362:
2293:
2213:
2187:
2159:
2077:
2020:
1987:
1967:
1947:
1924:
1834:
1814:
1790:
1761:
1687:
1616:
1583:
1550:
1493:
1439:
1383:
1320:
1258:
1178:, and is called a
1161:
1099:
1025:
989:
807:
685:
616:
585:
528:
498:
418:
265:
202:
137:
103:stochastic process
3479:Symbolic dynamics
3343:{\displaystyle K}
3259:{\displaystyle K}
3230:978-1-84800-047-6
3061:Loosely Bernoulli
3016:Sinai's billiards
2968:symbolic dynamics
2964:dynamical systems
2880:
2715:Cartesian product
2531:, as this is the
2216:{\displaystyle G}
1990:{\displaystyle B}
1970:{\displaystyle A}
1950:{\displaystyle M}
1922:
1862:
1837:{\displaystyle B}
1817:{\displaystyle A}
1788:
1318:
1277:Bernoulli process
1120:. The quadruplet
695:, its measure is
437:Bernoulli measure
52:dynamical systems
48:symbolic dynamics
44:Bernoulli process
16:(Redirected from
3486:
3453:
3439:
3433:
3432:
3430:
3410:
3404:
3403:
3401:
3368:
3362:
3361:
3349:
3347:
3346:
3341:
3325:
3319:
3312:
3306:
3296:
3290:
3289:
3287:
3265:
3263:
3262:
3257:
3241:
3235:
3234:
3216:
3210:
3209:
3192:
3186:
3185:
3183:
3158:Ornstein, Donald
3154:
3145:
3140:Pierre Gaspard,
3138:
3132:
3118:
3112:
3105:
2958:
2956:
2955:
2950:
2942:
2919:
2904:
2903:
2892:
2873:
2872:
2871:
2848:
2846:
2845:
2840:
2829:
2804:
2802:
2801:
2796:
2788:
2769:relative entropy
2762:
2760:
2759:
2754:
2743:
2742:
2709:
2707:
2706:
2701:
2696:
2695:
2680:
2679:
2669:
2664:
2612:
2610:
2609:
2604:
2555:is taken as the
2554:
2552:
2551:
2546:
2530:
2528:
2527:
2524:{\displaystyle }
2522:
2498:
2496:
2495:
2490:
2488:
2487:
2471:
2469:
2468:
2463:
2439:
2437:
2436:
2431:
2429:
2428:
2406:
2404:
2403:
2398:
2371:
2369:
2368:
2363:
2355:
2354:
2302:
2300:
2299:
2294:
2292:
2291:
2276:
2275:
2248:
2247:
2222:
2220:
2219:
2214:
2196:
2194:
2193:
2188:
2186:
2168:
2166:
2165:
2160:
2158:
2157:
2156:
2140:
2139:
2112:
2111:
2086:
2084:
2083:
2078:
2067:
2066:
2029:
2027:
2026:
2021:
1996:
1994:
1993:
1988:
1976:
1974:
1973:
1968:
1956:
1954:
1953:
1948:
1933:
1931:
1930:
1925:
1923:
1921:
1910:
1909:
1901:
1889:
1863:
1855:
1843:
1841:
1840:
1835:
1823:
1821:
1820:
1815:
1799:
1797:
1796:
1791:
1789:
1781:
1770:
1768:
1767:
1762:
1751:
1750:
1732:
1731:
1719:
1718:
1696:
1694:
1693:
1688:
1680:
1679:
1661:
1660:
1648:
1647:
1625:
1623:
1622:
1617:
1612:
1611:
1592:
1590:
1589:
1584:
1579:
1578:
1559:
1557:
1556:
1551:
1546:
1545:
1544:
1543:
1526:
1525:
1524:
1523:
1502:
1500:
1499:
1494:
1489:
1488:
1476:
1475:
1448:
1446:
1445:
1440:
1438:
1437:
1425:
1424:
1415:
1414:
1392:
1390:
1389:
1384:
1382:
1381:
1369:
1368:
1359:
1358:
1329:
1327:
1326:
1321:
1319:
1311:
1301:Hamming distance
1285:adjacency matrix
1267:
1265:
1264:
1259:
1251:
1250:
1232:
1231:
1180:Bernoulli scheme
1170:
1168:
1167:
1162:
1145:
1144:
1108:
1106:
1105:
1100:
1095:
1094:
1073:
1072:
1039:dynamical system
1034:
1032:
1031:
1026:
1021:
1020:
998:
996:
995:
990:
985:
984:
972:
971:
953:
952:
940:
939:
927:
926:
914:
913:
901:
890:
886:
882:
881:
863:
862:
850:
849:
816:
814:
813:
808:
806:
805:
804:
803:
788:
783:
765:
761:
757:
756:
738:
737:
725:
724:
694:
692:
691:
688:{\displaystyle }
686:
681:
680:
662:
661:
649:
648:
625:
623:
622:
617:
615:
614:
594:
592:
591:
586:
575:
574:
537:
535:
534:
529:
527:
526:
507:
505:
504:
499:
497:
496:
495:
485:
484:
466:
465:
427:
425:
424:
419:
411:
372:
371:
350:
349:
337:
336:
324:
323:
274:
272:
271:
266:
264:
263:
262:
211:
209:
208:
203:
195:
194:
184:
179:
146:
144:
143:
138:
136:
135:
72:Markov partition
36:Bernoulli scheme
21:
3494:
3493:
3489:
3488:
3487:
3485:
3484:
3483:
3459:
3458:
3457:
3456:
3440:
3436:
3411:
3407:
3376:Weiss, Benjamin
3369:
3365:
3335:
3332:
3331:
3326:
3322:
3313:
3309:
3297:
3293:
3251:
3248:
3247:
3242:
3238:
3231:
3217:
3213:
3193:
3189:
3155:
3148:
3139:
3135:
3119:
3115:
3106:
3102:
3097:
3075:
3063:
3055:Bernoulli shift
3037:An invertible,
3035:
2977:
2935:
2912:
2896:
2885:
2884:
2864:
2863:
2859:
2857:
2854:
2853:
2822:
2814:
2811:
2810:
2781:
2779:
2776:
2775:
2738:
2737:
2726:
2723:
2722:
2691:
2687:
2675:
2671:
2665:
2654:
2639:
2636:
2635:
2622:
2567:
2564:
2563:
2540:
2537:
2536:
2535:). The measure
2504:
2501:
2500:
2483:
2479:
2477:
2474:
2473:
2445:
2442:
2441:
2424:
2420:
2412:
2409:
2408:
2380:
2377:
2376:
2347:
2343:
2317:
2314:
2313:
2287:
2283:
2271:
2270:
2243:
2242:
2231:
2228:
2227:
2208:
2205:
2204:
2182:
2180:
2177:
2176:
2152:
2151:
2147:
2135:
2134:
2107:
2106:
2095:
2092:
2091:
2062:
2061:
2050:
2047:
2046:
2036:
2034:Generalizations
2006:
2003:
2002:
1982:
1979:
1978:
1962:
1959:
1958:
1942:
1939:
1938:
1911:
1905:
1897:
1890:
1888:
1854:
1852:
1849:
1848:
1829:
1826:
1825:
1809:
1806:
1805:
1780:
1778:
1775:
1774:
1746:
1742:
1727:
1723:
1714:
1710:
1702:
1699:
1698:
1675:
1671:
1656:
1652:
1643:
1639:
1631:
1628:
1627:
1607:
1603:
1598:
1595:
1594:
1574:
1570:
1565:
1562:
1561:
1539:
1535:
1534:
1530:
1519:
1515:
1514:
1510:
1508:
1505:
1504:
1484:
1480:
1471:
1467:
1462:
1459:
1458:
1433:
1429:
1420:
1416:
1410:
1406:
1398:
1395:
1394:
1377:
1373:
1364:
1360:
1354:
1350:
1342:
1339:
1338:
1310:
1308:
1305:
1304:
1297:
1246:
1242:
1227:
1223:
1194:
1191:
1190:
1184:Bernoulli shift
1140:
1139:
1128:
1125:
1124:
1084:
1080:
1068:
1064:
1056:
1053:
1052:
1016:
1012:
1007:
1004:
1003:
980:
976:
967:
963:
948:
944:
935:
931:
922:
918:
909:
905:
894:
877:
873:
858:
854:
845:
841:
837:
833:
828:
825:
824:
799:
795:
794:
790:
784:
773:
752:
748:
733:
729:
720:
716:
712:
708:
703:
700:
699:
676:
672:
657:
653:
644:
640:
635:
632:
631:
610:
609:
607:
604:
603:
570:
569:
558:
555:
554:
522:
521:
519:
516:
515:
491:
490:
486:
480:
476:
461:
457:
446:
443:
442:
431:The associated
407:
367:
363:
345:
341:
332:
328:
316:
312:
286:
283:
282:
258:
257:
253:
227:
224:
223:
190:
186:
180:
169:
163:
160:
159:
131:
127:
125:
122:
121:
110:random variable
96:
68:smooth manifold
56:Axiom A systems
40:Bernoulli shift
28:
23:
22:
15:
12:
11:
5:
3492:
3482:
3481:
3476:
3474:Ergodic theory
3471:
3455:
3454:
3434:
3405:
3363:
3339:
3320:
3307:
3305:, pp. 768–771.
3291:
3255:
3236:
3229:
3211:
3187:
3146:
3133:
3113:
3099:
3098:
3096:
3093:
3092:
3091:
3086:
3081:
3074:
3071:
3062:
3059:
3034:
3031:
3027:amenable group
2976:
2973:
2960:
2959:
2948:
2945:
2941:
2938:
2934:
2931:
2928:
2925:
2922:
2918:
2915:
2911:
2908:
2902:
2899:
2895:
2891:
2888:
2883:
2879:
2876:
2870:
2867:
2862:
2838:
2835:
2832:
2828:
2825:
2821:
2818:
2794:
2791:
2787:
2784:
2752:
2749:
2746:
2741:
2736:
2733:
2730:
2711:
2710:
2699:
2694:
2690:
2686:
2683:
2678:
2674:
2668:
2663:
2660:
2657:
2653:
2649:
2646:
2643:
2621:
2618:
2614:
2613:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2544:
2520:
2517:
2514:
2511:
2508:
2486:
2482:
2461:
2458:
2455:
2452:
2449:
2427:
2423:
2419:
2416:
2396:
2393:
2390:
2387:
2384:
2373:
2372:
2361:
2358:
2353:
2350:
2346:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2304:
2303:
2290:
2286:
2282:
2279:
2274:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2246:
2241:
2238:
2235:
2212:
2202:discrete group
2185:
2170:
2169:
2155:
2150:
2146:
2143:
2138:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2110:
2105:
2102:
2099:
2076:
2073:
2070:
2065:
2060:
2057:
2054:
2040:direct product
2035:
2032:
2019:
2016:
2013:
2010:
1986:
1966:
1946:
1935:
1934:
1920:
1917:
1914:
1908:
1904:
1900:
1896:
1893:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1861:
1858:
1833:
1813:
1787:
1784:
1760:
1757:
1754:
1749:
1745:
1741:
1738:
1735:
1730:
1726:
1722:
1717:
1713:
1709:
1706:
1686:
1683:
1678:
1674:
1670:
1667:
1664:
1659:
1655:
1651:
1646:
1642:
1638:
1635:
1615:
1610:
1606:
1602:
1582:
1577:
1573:
1569:
1549:
1542:
1538:
1533:
1529:
1522:
1518:
1513:
1492:
1487:
1483:
1479:
1474:
1470:
1466:
1453:is a sequence
1436:
1432:
1428:
1423:
1419:
1413:
1409:
1405:
1402:
1380:
1376:
1372:
1367:
1363:
1357:
1353:
1349:
1346:
1332:string matches
1317:
1314:
1296:
1293:
1269:
1268:
1257:
1254:
1249:
1245:
1241:
1238:
1235:
1230:
1226:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1172:
1171:
1160:
1157:
1154:
1151:
1148:
1143:
1138:
1135:
1132:
1110:
1109:
1098:
1093:
1090:
1087:
1083:
1079:
1076:
1071:
1067:
1063:
1060:
1043:shift operator
1024:
1019:
1015:
1011:
1000:
999:
988:
983:
979:
975:
970:
966:
962:
959:
956:
951:
947:
943:
938:
934:
930:
925:
921:
917:
912:
908:
904:
900:
897:
893:
889:
885:
880:
876:
872:
869:
866:
861:
857:
853:
848:
844:
840:
836:
832:
818:
817:
802:
798:
793:
787:
782:
779:
776:
772:
768:
764:
760:
755:
751:
747:
744:
741:
736:
732:
728:
723:
719:
715:
711:
707:
684:
679:
675:
671:
668:
665:
660:
656:
652:
647:
643:
639:
613:
602:. A basis of
596:
595:
584:
581:
578:
573:
568:
565:
562:
544:direct product
525:
513:σ-algebra
509:
508:
494:
489:
483:
479:
475:
472:
469:
464:
460:
456:
453:
450:
435:is called the
429:
428:
417:
414:
410:
406:
403:
400:
396:
393:
390:
387:
384:
381:
378:
375:
370:
366:
362:
359:
356:
353:
348:
344:
340:
335:
331:
327:
322:
319:
315:
311:
308:
305:
302:
299:
296:
293:
290:
276:
275:
261:
256:
252:
249:
246:
243:
240:
237:
234:
231:
213:
212:
201:
198:
193:
189:
183:
178:
175:
172:
168:
134:
130:
95:
92:
80:shift operator
26:
9:
6:
4:
3:
2:
3491:
3480:
3477:
3475:
3472:
3470:
3469:Markov models
3467:
3466:
3464:
3452:
3451:0-387-90599-5
3448:
3444:
3438:
3429:
3424:
3420:
3416:
3409:
3400:
3395:
3391:
3387:
3386:
3381:
3377:
3373:
3367:
3359:
3355:
3351:
3337:
3324:
3317:
3311:
3304:
3301:
3295:
3286:
3281:
3277:
3273:
3272:
3267:
3253:
3240:
3232:
3226:
3222:
3215:
3208:
3204:
3203:
3198:
3191:
3182:
3177:
3173:
3169:
3168:
3163:
3159:
3153:
3151:
3143:
3137:
3131:
3130:0-19-853390-X
3127:
3123:
3117:
3110:
3104:
3100:
3090:
3087:
3085:
3082:
3080:
3077:
3076:
3070:
3068:
3058:
3056:
3052:
3048:
3044:
3040:
3030:
3028:
3024:
3019:
3017:
3013:
3009:
3008:Markov chains
3005:
3001:
2997:
2992:
2990:
2986:
2982:
2972:
2969:
2965:
2946:
2939:
2936:
2929:
2926:
2923:
2916:
2913:
2906:
2900:
2897:
2893:
2889:
2886:
2881:
2877:
2874:
2868:
2865:
2860:
2852:
2851:
2850:
2836:
2833:
2826:
2823:
2816:
2808:
2792:
2789:
2785:
2782:
2774:
2770:
2766:
2747:
2744:
2734:
2731:
2720:
2716:
2697:
2692:
2688:
2684:
2681:
2676:
2672:
2666:
2661:
2658:
2655:
2651:
2647:
2644:
2641:
2634:
2633:
2632:
2630:
2626:
2617:
2599:
2593:
2587:
2584:
2578:
2575:
2569:
2562:
2561:
2560:
2558:
2542:
2534:
2515:
2509:
2484:
2480:
2459:
2453:
2450:
2447:
2425:
2421:
2417:
2414:
2394:
2391:
2388:
2385:
2382:
2356:
2351:
2348:
2344:
2337:
2334:
2328:
2322:
2319:
2312:
2311:
2310:
2309:
2288:
2280:
2277:
2267:
2264:
2258:
2252:
2249:
2239:
2236:
2226:
2225:
2224:
2210:
2203:
2200:
2173:
2144:
2141:
2131:
2128:
2122:
2116:
2113:
2103:
2100:
2090:
2089:
2088:
2071:
2068:
2058:
2055:
2045:
2041:
2031:
2017:
2014:
2011:
2008:
2000:
1984:
1964:
1944:
1918:
1915:
1912:
1902:
1891:
1885:
1882:
1879:
1873:
1870:
1867:
1856:
1847:
1846:
1845:
1831:
1811:
1803:
1782:
1771:
1758:
1755:
1752:
1747:
1743:
1739:
1736:
1733:
1728:
1724:
1720:
1715:
1711:
1707:
1704:
1697:and likewise
1684:
1681:
1676:
1672:
1668:
1665:
1662:
1657:
1653:
1649:
1644:
1640:
1636:
1633:
1626:are ordered:
1608:
1604:
1575:
1571:
1547:
1540:
1536:
1531:
1527:
1520:
1516:
1511:
1485:
1481:
1477:
1472:
1468:
1456:
1452:
1434:
1430:
1426:
1421:
1417:
1411:
1407:
1403:
1400:
1378:
1374:
1370:
1365:
1361:
1355:
1351:
1347:
1344:
1335:
1333:
1312:
1302:
1292:
1290:
1286:
1282:
1278:
1274:
1255:
1247:
1243:
1239:
1236:
1233:
1228:
1224:
1217:
1214:
1211:
1205:
1199:
1196:
1189:
1188:
1187:
1185:
1181:
1177:
1155:
1152:
1149:
1146:
1136:
1133:
1123:
1122:
1121:
1119:
1115:
1096:
1091:
1088:
1085:
1081:
1077:
1069:
1065:
1058:
1051:
1050:
1049:
1047:
1044:
1040:
1035:
1017:
1013:
981:
977:
973:
968:
964:
960:
957:
954:
949:
945:
941:
936:
932:
928:
923:
919:
915:
910:
906:
891:
887:
878:
874:
870:
867:
864:
859:
855:
851:
846:
842:
834:
830:
823:
822:
821:
800:
796:
791:
785:
780:
777:
774:
770:
766:
762:
753:
749:
745:
742:
739:
734:
730:
726:
721:
717:
709:
705:
698:
697:
696:
677:
673:
669:
666:
663:
658:
654:
650:
645:
641:
629:
628:cylinder sets
601:
600:measure space
579:
576:
566:
563:
553:
552:
551:
549:
545:
541:
514:
481:
477:
473:
470:
467:
462:
458:
451:
448:
441:
440:
439:
438:
434:
415:
404:
401:
391:
388:
385:
382:
379:
373:
368:
364:
360:
354:
351:
346:
342:
338:
333:
329:
325:
320:
317:
313:
309:
306:
300:
297:
291:
288:
281:
280:
279:
250:
247:
244:
241:
238:
232:
229:
222:
221:
220:
218:
199:
196:
191:
187:
181:
176:
173:
170:
166:
158:
157:
156:
154:
150:
132:
128:
119:
115:
111:
108:
104:
101:
100:discrete-time
91:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
3442:
3437:
3418:
3414:
3408:
3389:
3383:
3366:
3360:: 4263–4280.
3357:
3353:
3323:
3310:
3302:
3299:
3294:
3278:(1): 16–38.
3275:
3269:
3239:
3220:
3214:
3200:
3190:
3171:
3165:
3141:
3136:
3121:
3116:
3108:
3107:P. Shields,
3103:
3084:Markov chain
3064:
3046:
3036:
3022:
3020:
3012:Anosov flows
2993:
2978:
2961:
2809:, such that
2806:
2805:of the base
2764:
2712:
2623:
2615:
2557:Haar measure
2374:
2308:group action
2305:
2174:
2171:
2037:
1936:
1801:
1772:
1454:
1450:
1336:
1331:
1298:
1281:Markov shift
1272:
1270:
1183:
1179:
1173:
1113:
1111:
1045:
1036:
1001:
819:
597:
547:
539:
510:
436:
430:
277:
217:sample space
214:
152:
148:
117:
113:
97:
75:
58:) exhibit a
39:
35:
29:
3174:: 337–352.
2223:, so that
107:independent
105:where each
74:. The term
32:mathematics
3463:Categories
3095:References
3051:isomorphic
2985:isomorphic
2620:Properties
2001:only when
94:Definition
90:is equal.
64:Cantor set
3428:1103.4424
3421:: 67–78.
3392:: 1–141.
3207:EMS Press
3049:if it is
3006:, finite
2930:ν
2927:
2907:ν
2894:∈
2882:∑
2878:−
2817:ν
2790:⊂
2773:partition
2748:ν
2685:
2652:∑
2648:−
2625:Ya. Sinai
2588:μ
2570:μ
2543:μ
2513:→
2457:→
2418:∈
2392:∈
2349:−
2281:ν
2253:μ
2199:countable
2145:ν
2117:μ
2072:ν
1886:−
1860:¯
1786:¯
1753:≤
1737:⋯
1708:≤
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3160:(1970).
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626:is the
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147:, with
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