Relative pressure, relative equilibrium states, compensation functions and many-to-one codes between subshifts

Author:
Peter Walters

Journal:
Trans. Amer. Math. Soc. **296** (1986), 1-31

MSC:
Primary 28D99; Secondary 58F11

MathSciNet review:
837796

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be continuous maps of compact metrizable spaces, and let be a continuous surjection with . We investigate the notion of relative pressure, which was introduced by Ledrappier and Walters, and study some maximal relative pressure functions that tie in with relative equilibrium states. These ideas are connected with the notion of compensation function, first considered by Boyle and Tuncel, and we show that a compensation function always exists when and are subshifts. A function is a compensation function if . When and are topologically mixing subshifts of finite type, we relate compensation functions to lifting -invariant measures to -invariant measures, obtaining some results of Boyle and Tuncel. We use compensation functions to describe different types of quotient maps . An example is given where no compensation function exists.

**[**L. M. Abramov and V. A. Rohlin,**A,R**]*Entropy of a skew product of mappings with invariant measure*, Vestnik Leningrad. Univ.**17**(1962), no. 7, 5–13 (Russian, with English summary). MR**0140660****[**Rufus Bowen,**B**]*Equilibrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR**0442989****[**Mike Boyle and Selim Tuncel,**B,T**]*Infinite-to-one codes and Markov measures*, Trans. Amer. Math. Soc.**285**(1984), no. 2, 657–684. MR**752497**, 10.1090/S0002-9947-1984-0752497-0**[**G. A. Hedlund,**H**]*Endomorphisms and automorphisms of the shift dynamical system*, Math. Systems Theory**3**(1969), 320–375. MR**0259881****[**Robert B. Israel,**I**]*Convexity in the theory of lattice gases*, Princeton University Press, Princeton, N.J., 1979. Princeton Series in Physics; With an introduction by Arthur S. Wightman. MR**517873****[**F. Ledrappier,**L**]*Principe variational et systèmes symboliques*, Comment. Math. Phys.**33**(1973), 119-128.**[**François Ledrappier and Peter Walters,**L,W**]*A relativised variational principle for continuous transformations*, J. London Math. Soc. (2)**16**(1977), no. 3, 568–576. MR**0476995****[**Brian Marcus, Karl Petersen, and Susan Williams,**M,P,W**]*Transmission rates and factors of Markov chains*, Conference in modern analysis and probability (New Haven, Conn., 1982), Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 279–293. MR**737408**, 10.1090/conm/026/737408**[**Nelson G. Markley and Michael E. Paul,**M,P**]*Equilibrium states of grid functions*, Trans. Amer. Math. Soc.**274**(1982), no. 1, 169–191. MR**670926**, 10.1090/S0002-9947-1982-0670926-6**[**William Parry and Selim Tuncel,**P,T**]*Classification problems in ergodic theory*, London Mathematical Society Lecture Note Series, vol. 67, Cambridge University Press, Cambridge-New York, 1982. Statistics: Textbooks and Monographs, 41. MR**666871****[**David Ruelle,**R**]*Thermodynamic formalism*, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR**511655****[**Selim Tuncel,**T**]*Conditional pressure and coding*, Israel J. Math.**39**(1981), no. 1-2, 101–112. MR**617293**, 10.1007/BF02762856**[**Peter Walters,**W**]*Ruelle’s operator theorem and 𝑔-measures*, Trans. Amer. Math. Soc.**214**(1975), 375–387. MR**0412389**, 10.1090/S0002-9947-1975-0412389-8**[**Peter Walters,**W**]*Invariant measures and equilibrium states for some mappings which expand distances*, Trans. Amer. Math. Soc.**236**(1978), 121–153. MR**0466493**, 10.1090/S0002-9947-1978-0466493-1**[**Peter Walters,**W**]*An introduction to ergodic theory*, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR**648108**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
28D99,
58F11

Retrieve articles in all journals with MSC: 28D99, 58F11

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1986-0837796-8

Keywords:
Pressure,
compensation function,
equilibrium state,
Markov measure,
code

Article copyright:
© Copyright 1986
American Mathematical Society