March  2015, 35(3): 1163-1177. doi: 10.3934/dcds.2015.35.1163

Robust transitivity of maps of the real line

1. 

Universidad Centroccidental Lisandro Alvarado, Decanato de Ciencias y Tecnología, Barquisimeto, Venezuela

Received  February 2014 Revised  March 2014 Published  October 2014

In the set of continuously differentiable maps of the real line with a discontinuity, equipped with the uniform topology, it will be shown that the subset of transitive ones has nonempty interior.
Citation: Sergio Muñoz. Robust transitivity of maps of the real line. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1163-1177. doi: 10.3934/dcds.2015.35.1163
References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

R. Adler, F-expansions revisited, Lecture Notes in Math., 318 (1973), 1-5.  Google Scholar

[3]

R. Adler and B. Weiss, The ergodic infinite measure preserving transformation of Boole, Israel Journal of Math, 16 (1973), 263-278. doi: 10.1007/BF02756706.  Google Scholar

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D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp.  Google Scholar

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C. Bonatti, L. Díaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math., 158 (2003), 355-418. doi: 10.4007/annals.2003.158.355.  Google Scholar

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G. Boole, On the comparison of transcendents with certain applications to the theory of definite integrals, Philos. Trans. Roy. Soc. London, 8 (1856), 461-463. doi: 10.1098/rspl.1856.0122.  Google Scholar

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L. Díaz, E. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43. doi: 10.1007/BF02392945.  Google Scholar

[8]

R. Devaney, The baker transformation and a mapping associated to the restricted three-body problem, Comm. Math. Phys., 80 (1981), 465-476. doi: 10.1007/BF01941657.  Google Scholar

[9]

M. V. Jakobson, On Smooth mappings of the circle into itself, Math. USSR Sb, 85 (1971), 163-188.  Google Scholar

[10]

R. Mañe, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8.  Google Scholar

[11]

P. Mendes, On Anosov diffeomorphisms on the plane, Proc. Amer. Math. Soc., 63 (1977), 231-235. doi: 10.1090/S0002-9939-1977-0461585-X.  Google Scholar

[12]

M. Peixoto, On structural stability, Ann. of Math., 69 (1959), 199-222. doi: 10.2307/1970100.  Google Scholar

[13]

E. Pujals, From hyperbolicity to dominated splitting, Fields Institute Communications, 51 (2007), 89-102.  Google Scholar

[14]

F. Schweiger, Numbertheoretical endomorphisms with $\sigma$-finite invariant measure, Israel Journal of Math., 21 (1975), 308-318. doi: 10.1007/BF02757992.  Google Scholar

[15]

F. Schweiger, tan ($x$) is ergodic, Proceedings of the American Mathematical Society, 71 (1978), 54-56.  Google Scholar

[16]

F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.  Google Scholar

[17]

M. Shub, Topologically Transitive Diffeomorphisms of $T^4$, Lecture Notes in Math., 206. Springer Verlag, Berlin-New York, 1971. Google Scholar

[18]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed point, Israel Journal of Math., 37 (1980), 303-314. doi: 10.1007/BF02788928.  Google Scholar

[19]

M. Thaler, Transformations on [0,1] with infinite invariant measures, Israel Journal of Math., 46 (1983), 67-96. doi: 10.1007/BF02760623.  Google Scholar

[20]

T.-Y. Li and F. Schweiger, The generalized transformation of Boole is ergodic, Manuscripta Math., 25 (1978), 161-167. doi: 10.1007/BF01168607.  Google Scholar

[21]

R. Zweimüller, Ergodic structure and invariant densities of non-Markoviant interval maps with indifferent fixed points, Nonlinearity, 11 (1998), 1263-1276. doi: 10.1088/0951-7715/11/5/005.  Google Scholar

[22]

R. Zweimüller, Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergodic Theory & Dynamical Systems, 20 (2000), 1519-1549. doi: 10.1017/S0143385700000821.  Google Scholar

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

R. Adler, F-expansions revisited, Lecture Notes in Math., 318 (1973), 1-5.  Google Scholar

[3]

R. Adler and B. Weiss, The ergodic infinite measure preserving transformation of Boole, Israel Journal of Math, 16 (1973), 263-278. doi: 10.1007/BF02756706.  Google Scholar

[4]

D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp.  Google Scholar

[5]

C. Bonatti, L. Díaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math., 158 (2003), 355-418. doi: 10.4007/annals.2003.158.355.  Google Scholar

[6]

G. Boole, On the comparison of transcendents with certain applications to the theory of definite integrals, Philos. Trans. Roy. Soc. London, 8 (1856), 461-463. doi: 10.1098/rspl.1856.0122.  Google Scholar

[7]

L. Díaz, E. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43. doi: 10.1007/BF02392945.  Google Scholar

[8]

R. Devaney, The baker transformation and a mapping associated to the restricted three-body problem, Comm. Math. Phys., 80 (1981), 465-476. doi: 10.1007/BF01941657.  Google Scholar

[9]

M. V. Jakobson, On Smooth mappings of the circle into itself, Math. USSR Sb, 85 (1971), 163-188.  Google Scholar

[10]

R. Mañe, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8.  Google Scholar

[11]

P. Mendes, On Anosov diffeomorphisms on the plane, Proc. Amer. Math. Soc., 63 (1977), 231-235. doi: 10.1090/S0002-9939-1977-0461585-X.  Google Scholar

[12]

M. Peixoto, On structural stability, Ann. of Math., 69 (1959), 199-222. doi: 10.2307/1970100.  Google Scholar

[13]

E. Pujals, From hyperbolicity to dominated splitting, Fields Institute Communications, 51 (2007), 89-102.  Google Scholar

[14]

F. Schweiger, Numbertheoretical endomorphisms with $\sigma$-finite invariant measure, Israel Journal of Math., 21 (1975), 308-318. doi: 10.1007/BF02757992.  Google Scholar

[15]

F. Schweiger, tan ($x$) is ergodic, Proceedings of the American Mathematical Society, 71 (1978), 54-56.  Google Scholar

[16]

F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.  Google Scholar

[17]

M. Shub, Topologically Transitive Diffeomorphisms of $T^4$, Lecture Notes in Math., 206. Springer Verlag, Berlin-New York, 1971. Google Scholar

[18]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed point, Israel Journal of Math., 37 (1980), 303-314. doi: 10.1007/BF02788928.  Google Scholar

[19]

M. Thaler, Transformations on [0,1] with infinite invariant measures, Israel Journal of Math., 46 (1983), 67-96. doi: 10.1007/BF02760623.  Google Scholar

[20]

T.-Y. Li and F. Schweiger, The generalized transformation of Boole is ergodic, Manuscripta Math., 25 (1978), 161-167. doi: 10.1007/BF01168607.  Google Scholar

[21]

R. Zweimüller, Ergodic structure and invariant densities of non-Markoviant interval maps with indifferent fixed points, Nonlinearity, 11 (1998), 1263-1276. doi: 10.1088/0951-7715/11/5/005.  Google Scholar

[22]

R. Zweimüller, Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergodic Theory & Dynamical Systems, 20 (2000), 1519-1549. doi: 10.1017/S0143385700000821.  Google Scholar

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