Giulio Tiozzo - Papers

Complex dynamics

1. Master teapots and entropy algorithms for the Mandelbrot set. [PDF],
   (with K. Lindsey, C. Wu), Transactions of the American Mathematical Society, accepted (2024). See also the Gallery.
   
2. Metrics on trees I. The tower algorithm for interval maps [PDF],
   arXiv:2112.02398.
3. The core entropy for polynomials of higher degree [PDF] [Journal],
   (with Y. Gao), Journal of the European Mathematical Society 24 (2022), no. 7, 2555-2603.
4. Continuity of core entropy of quadratic polynomials [PDF] [Journal],
   Inventiones Mathematicae 203 (2016), no. 3, 891-921.
5. Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set [PDF],
   Advances in Mathematics 273 (2015), 651-715.
6. Generalizations of Douady's magic formula [PDF] [Journal],
   (with A. Epstein), Ergodic Theory Dynam. Systems 42 (2022), no. 9, 2784-2799.
7. A central limit theorem for the degree of a random product of Cremona transformations [PDF],
   (with N.-B. Dang), Indiana Univ. Math. J. 72 (2023), no. 1, 301-329.

Real dynamics

8. Thermodynamic formalism for coarse expanding dynamical systems [PDF],
   (with T. Das, F. Przytycki, M. Urbański, A. Zdunik), Communications in Mathematical Physics 384 (2021), 165-199.
9. The local Hölder exponent for the entropy of real unimodal maps [PDF],
   Science China Mathematics 61 (2018), no. 12, 2299-2310 (special issue in honor of Tan Lei).
10. Galois conjugates of entropies of real unimodal maps [PDF],
    International Mathematics Research Notices 2020 (2020), no. 2, 607-640. Also check out the related Gallery.
11. The local Hölder exponent for the dimension of invariant subsets of the circle [PDF],
    (with C. Carminati), Ergodic Theory and Dynamical Systems 37 (2017), no. 6, 1825-1840.
12. Dynamics of continued fractions and kneading sequences of unimodal maps [PDF],
    (with C. Bonanno, C. Carminati, S. Isola), Discrete and Continuous Dynamical Systems 33, no. 4 (2013), 1313-1332.

Random walks on groups

13. Roots of Alexander polynomials of random positive 3-braids [PDF],
    (with N. Dunfield), available at arXiv:2402.06771.
14. Genericity of contracting geodesics in groups [PDF],
    (with K. Chawla, I. Choi), available at arXiv:2308.01877.
15. Equidistribution of hyperbolic groups in homogeneous spaces [PDF], [Journal]
    (with I. Gekhtman, S. Taylor), Mathematische Annalen, accepted (2024).
16. The Poisson boundary of hyperbolic groups without moment conditions [PDF],
    (with K. Chawla, B. Forghani, J. Frisch), available at arXiv:2209.02114.
17. A global shadow lemma and logarithm law for geometrically finite Hilbert geometries [PDF],
    (with H. Bray), available at arXiv:2111.04618.
18. The fundamental inequality for cocompact Fuchsian groups [PDF],
    (with P. Kosenko), Forum of Math. Sigma 10 (2022), E102.
19. Central limit theorems for counting measures in coarse negative curvature [PDF] [slides] [video],
    (with I. Gekhtman, S. Taylor), Compositio Mathematica 158 (2022), no. 10, 1980-2013.
20. Sublinearly Morse Boundary II: Proper geodesic spaces [PDF],
    (with Y. Qing, K. Rafi), Geometry and Topology 28 (2024), no. 4, 1829-1889.
21. Appendix in Sublinearly Morse boundary I: CAT(0) spaces [PDF],
    (by Y. Qing, K. Rafi), Advances in Mathematics 404 (2022), 108442.
22. Cusp excursion in hyperbolic manifolds and singularity of harmonic measure [PDF],
    (with A. Randecker), Journal of Modern Dynamics 17 (2021), 183-211.
23. Entropy and drift for Gibbs measures on geometrically finite manifolds [PDF],
    (with I. Gekhtman), Transactions of the American Mathematical Society 373 (2020), 2949-2980.
24. Shannon's theorem for locally compact groups [PDF] [slides],
    (with B. Forghani), Annals of Probability 50 (2022), no. 1, 61-89.
25. Excursions of generic geodesics in right-angled Artin groups and graph products [PDF],
    (with Y. Qing), International Mathematics Research Notices (2019), rnz294.
26. A central limit theorem for random closed geodesics: proof of the Chas-Li-Maskit conjecture [PDF],
    (with I. Gekhtman, S. Taylor), Advances in Mathematics 358 (2019), 106852.
27. Random walks, WPD actions, and the Cremona group [PDF] [Journal],
    (with J. Maher), Proceedings of the London Mathematical Society 123 (2021), no. 2, 153-202.
28. Random walks of infinite moment on free semigroups [PDF],
    (with B. Forghani), Probability Theory and Related Fields 175 (2019), no. 3, 1099-1122.
29. Counting problems in graph products and relatively hyperbolic groups [PDF],
    (with I. Gekhtman, S. Taylor), Israel Journal of Mathematics 237 (2020), 311-371.
30. Counting loxodromics for hyperbolic actions [PDF],
    (with I. Gekhtman, S. Taylor), Journal of Topology 11 (2018), no. 2, 379-419.
31. Sublinear deviation between geodesics and sample paths [PDF],
    Duke Mathematical Journal 164 (2015), no. 3, 511-539.
32. Random extensions of free groups and surface groups are hyperbolic [PDF],
    (with S. Taylor), International Mathematics Research Notices 2016 (2016), no. 1, 294-310.
33. Random walks on weakly hyperbolic groups [PDF],
    (with J. Maher), Journal für die reine und angewandte Mathematik 742 (2018), 187-239.
34. Word length statistics of Teichmüller geodesics and singularity of harmonic measure [PDF],
    (with V. Gadre, J. Maher), Commentarii Mathematici Helvetici 92 (2017), no. 1, 1-36.
35. Word length statistics and Lyapunov exponents for Fuchsian groups with cusps [PDF],
    (with V. Gadre, J. Maher), New York Journal of Mathematics 21 (2015), 511-531.

Dynamics of continued fractions

36. Continued fractions with SL(2, Z)-branches: combinatorics and entropy [PDF],
    (with C. Carminati, S. Isola), Transactions of the American Mathematical Society 370 (2018), 4927-4973.
37. A canonical thickening of Q and the entropy of α-continued fraction transformations [PDF],
    (with C. Carminati), Ergodic Theory and Dynamical Systems 32 (2012), no. 4, 1249-1269.
38. The entropy of α-continued fractions: analytical results [PDF],
    Annali della Scuola Normale Superiore di Pisa 13 (2014), 1009-1037.
39. Tuning and plateaux for the entropy of α-continued fractions [PDF],
    (with C. Carminati), Nonlinearity 26 (2013), 1049-1070.
40. The entropy of α-continued fractions: numerical results [PDF],
    (with C. Carminati, S. Marmi, A. Profeti), Nonlinearity 23 (2010), 2429-2456.
41. The bifurcation locus for numbers of bounded type [PDF],
    (with C. Carminati), Ergodic Theory and Dynamical Systems 42 (2022), no. 7, 2239-2269.

Complex analysis

42. Generalised continuation by means of right limits [PDF],
    (with D. Sauzin), Journal d'Analyse Mathématique 133 (2017), no. 1, 27-49.