DYNAMICS, ERGODIC THEORY, AND GEOMETRY
CONTENTS PART I ERGODIC THEORY AND GEOMETRIC MEASURES GEOMETRIC MEASURE THEORY MEASURES, INTEGRALS, AND MEASURE SPACES MEASURES ON METRIC SPACES: (METRIC) OUTER MEASURES AND WEAK?CONVERGENCE COVERING THEOREMS: 4R, BESICOVITCH, AND VITALI TYPE; LEBESGUE DENSITY THEOREM CONDITIONAL EXPECTATIONS AND MARTINGALE THEOREMS HAUSDORFF AND PACKING MEASURES: HAUSDORFF AND PACKING DIMENSIONS HAUSDORFF AND PACKING MEASURES: FROSTMAN CONVERSE-TYPE THEOREMS HAUSDORFF AND PACKING DIMENSIONS OF MEASURES BOX-COUNTING DIMENSIONS INVARIANT MEASURES: FINITE AND IN?NITE QUASI-INVARIANT TRANSFORMATIONS: ERGODICITY AND CONSERVATIVITY INVARIANT MEASURES: FIRST RETURN MAP (INDUCING); POINCAR´E RECURRENCE THEOREM ERGODIC THEOREMS: BIRKHOFF, VON NEUMANN, AND HOPF ABSOLUTELY CONTINUOUS ?-FINITE INVARIANT MEASURES: MARCO MARTENS’S APPROACH PROBABILITY (FINITE) INVARIANT MEASURES: BASIC PROPERTIES AND EXISTENCE BASIC PROPERTIES OF PROBABILITY INVARIANT MEASURES EXISTENCE OF BOREL PROBABILITY INVARIANT MEASURES: BOGOLYUBOV–KRYLOV THEOREM EXAMPLES OF INVARIANT AND ERGODIC MEASURES PROBABILITY (FINITE) INVARIANT MEASURES: FINER PROPERTIES THE LAW OF THE ITERATED LOGARITHM DECAY OF CORRELATIONS AND THE CENTRAL LIMIT THEOREMS: LAI-SANG YOUNG TOWERS ROKHLIN NATURAL EXTENSION IN?NITE INVARIANT MEASURES: FINER PROPERTIES COUNTEREXAMPLES TO ERGODIC THEOREMS WEAK ERGODIC THEOREMS DARLING–KAC THEOREM: ABSTRACT VERSION POINTS OF IN?NITE CONDENSATION: ABSTRACT SETTING MEASURE-THEORETIC ENTROPY PARTITIONS INFORMATION AND CONDITIONAL INFORMATION FUNCTIONS ENTROPY AND CONDITIONAL ENTROPY FOR PARTITIONS ENTROPY OF A (PROBABILITY) MEASURE-PRESERVING ENDOMORPHISM SHANNON–MCMILLAN–BREIMAN THEOREM ABRAMOV’S FORMULA AND KRENGEL’S ENTROPY (IN?NITE MEASURES ALLOWED) THERMODYNAMIC FORMALISM TOPOLOGICAL PRESSURE BOWEN’S DE?NITION OF TOPOLOGICAL PRESSURE BASIC PROPERTIES OF TOPOLOGICAL PRESSURE EXAMPLES THE VARIATIONAL PRINCIPLE AND EQUILIBRIUM STATES P BLI H D LI B C B ID U I I P PART II COMPLEX ANALYSIS, CONFORMAL MEASURES, AND GRAPH DIRECTED MARKOV SYSTEMS SELECTED TOPICS FROM COMPLEX ANALYSIS RIEMANN SURFACES, NORMAL FAMILIES, AND MONTEL’S THEOREM EXTREMAL LENGTHS AND MODULI OF TOPOLOGICAL ANNULI KOEBE DISTORTION THEOREMS LOCAL PROPERTIES OF CRITICAL POINTS OF HOLOMORPHIC FUNCTIONS PROPER ANALYTIC MAPS AND THEIR DEGREE RIEMANN–HURWITZ FORMULA INVARIANT MEASURES FOR HOLOMORPHIC MAPS F IN A(X) OR IN AW(X) PRELIMINARIES THREE AUXILIARY PARTITIONS PESIN’S THEORY RUELLE’S INEQUALITY VOLUME LEMMAS AND HAUSDORFF DIMENSIONS OF INVARIANT MEASURES SULLIVAN CONFORMAL MEASURES FOR HOLOMORPHIC MAPS F IN A(X) AND IN AW(X) GENERAL CONCEPT OF CONFORMAL MEASURES RADIAL SUBSETS OF X, DYNAMICAL DIMENSIONS, AND SULLIVAN CONFORMAL MEASURES FOR HOLOMORPHIC MAPS F IN A(X) AND IN AW(X) CONFORMAL PAIRS OF MEASURES GRAPH DIRECTED MARKOV SYSTEMS SUBSHIFTS OF FINITE TYPE OVER IN?NITE ALPHABETS: TOPOLOGICAL PRESSURE CONFORMAL GRAPH DIRECTED MARKOV SYSTEMS TOPOLOGICAL PRESSURE, ?-NUMBER, AND BOWEN’S PARAMETER HAUSDORFF DIMENSIONS AND BOWEN’S FORMULA FOR GDMSS CONFORMAL AND INVARIANT MEASURES FOR CGDMSS FINER GEOMETRICAL PROPERTIES OF CGDMSS THE STRONG OPEN SET CONDITION CONFORMAL MAXIMAL GRAPH DIRECTED SYSTEMS 11 10 CONJUGACIES OF CONFORMAL GRAPH DIRECTED SYSTEMS NICE SETS FOR ANALYTIC MAPS PRE-NICE AND NICE SETS AND THE RESULTING CGMDS: PRELIMINARIES THE EXISTENCE OF PRE-NICE SETS THE EXISTENCE OF NICE SETS THE MAXIMAL GDMSS INDUCED BY NICE SETS AN AUXILIARY TECHNICAL RESULT REFERENCES