Free energy and equilibrium states for families of interval maps / Neil Dobbs, Mike Todd.

Format:

Book

Author/Creator:

Dobbs, Neil, author.

Todd, Mike (Professor of mathematics), author.

Series:

Memoirs of the American Mathematical Society ; 1417.

Memoirs of the American Mathematical Society, 0065-9266 ; 1417

Language:

English

Subjects (All):

Ergodic theory.

Dynamics.

Physical Description:

v, 103 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"We study continuity, and lack thereof, of thermodynamical properties for onedimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (including statistical stability). Metric entropy, not semicontinuous as a general multimodal map varies, is shown to be upper semicontinuous under an appropriate hypothesis on critical orbits. Equilibrium states vary continuously, under mild hypotheses, as one varies the parameter and the map. We give a general method for constructing induced maps which automatically give strong exponential tail estimates. This also allows us to recover, and further generalise, recent results concerning statistical properties (decay of correlations, etc.). Counterexamples to statistical stability are given which also show sharpness of the main results"-- Provided by publisher.

Contents:

Chapter 1. Introduction

Chapter 2. Topological structures

Chapter 3. Measures and entropy

Chapter 4. Light limit measures and upper-semicontinuity of metric entropy

Chapter 5. Non-positive Schwarzian derivative

Chapter 6. Almost upper-semicontinuity of the free energy

Chapter 7. Katok theory, pressure and exponential tails

Chapter 8. Instability for Collet-Eckmann maps

Chapter 9. Positive entropy does not imply statistical stability

Bibliography.

Notes:

Includes bibliographical references and index.

ISBN:

9781470461263

1470461269

OCLC:

1386970512