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

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- by Peter Walters PDF
- Trans. Amer. Math. Soc.
**296**(1986), 1-31 Request permission

## Abstract:

Let $S:X \to X,T:Y \to Y$ be continuous maps of compact metrizable spaces, and let $\pi :X \to Y$ be a continuous surjection with $\pi \circ S = T \circ \pi$. 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 $S$ and $T$ are subshifts. A function $F \in C(X)$ is a compensation function if $P(S,F + \phi \circ \pi ) = P(T,\phi )\forall \phi \in C(Y)$. When $S$ and $T$ are topologically mixing subshifts of finite type, we relate compensation functions to lifting $T$-invariant measures to $S$-invariant measures, obtaining some results of Boyle and Tuncel. We use compensation functions to describe different types of quotient maps $\pi$. An example is given where no compensation function exists.## References

- L. M. Abramov and V. A. Rohlin,
*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,
*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,
*Infinite-to-one codes and Markov measures*, Trans. Amer. Math. Soc.**285**(1984), no. 2, 657–684. MR**752497**, DOI 10.1090/S0002-9947-1984-0752497-0 - G. A. Hedlund,
*Endomorphisms and automorphisms of the shift dynamical system*, Math. Systems Theory**3**(1969), 320–375. MR**259881**, DOI 10.1007/BF01691062 - Robert B. Israel,
*Convexity in the theory of lattice gases*, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1979. With an introduction by Arthur S. Wightman. MR**517873**
F. Ledrappier, - François Ledrappier and Peter Walters,
*A relativised variational principle for continuous transformations*, J. London Math. Soc. (2)**16**(1977), no. 3, 568–576. MR**476995**, DOI 10.1112/jlms/s2-16.3.568 - Brian Marcus, Karl Petersen, and Susan Williams,
*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**, DOI 10.1090/conm/026/737408 - Nelson G. Markley and Michael E. Paul,
*Equilibrium states of grid functions*, Trans. Amer. Math. Soc.**274**(1982), no. 1, 169–191. MR**670926**, DOI 10.1090/S0002-9947-1982-0670926-6 - William Parry and Selim Tuncel,
*Classification problems in ergodic theory*, Statistics: Textbooks and Monographs, vol. 41, Cambridge University Press, Cambridge-New York, 1982. MR**666871** - David Ruelle,
*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,
*Conditional pressure and coding*, Israel J. Math.**39**(1981), no. 1-2, 101–112. MR**617293**, DOI 10.1007/BF02762856 - Peter Walters,
*Ruelle’s operator theorem and $g$-measures*, Trans. Amer. Math. Soc.**214**(1975), 375–387. MR**412389**, DOI 10.1090/S0002-9947-1975-0412389-8 - Peter Walters,
*Invariant measures and equilibrium states for some mappings which expand distances*, Trans. Amer. Math. Soc.**236**(1978), 121–153. MR**466493**, DOI 10.1090/S0002-9947-1978-0466493-1 - Peter Walters,
*An introduction to ergodic theory*, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR**648108**

*Principe variational et systèmes symboliques*, Comment. Math. Phys.

**33**(1973), 119-128.

## Additional Information

- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**296**(1986), 1-31 - MSC: Primary 28D99; Secondary 58F11
- DOI: https://doi.org/10.1090/S0002-9947-1986-0837796-8
- MathSciNet review: 837796